Content-Type: multipart/mixed; boundary="-------------0407291119751" This is a multi-part message in MIME format. ---------------0407291119751 Content-Type: text/plain; name="04-234.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-234.comments" Author's e-mail: zakowicz@physik.uni-giessen.de ---------------0407291119751 Content-Type: text/plain; name="04-234.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-234.keywords" Dirac equation, Volkov solutions, normalization, orthogonality ---------------0407291119751 Content-Type: application/postscript; name="Volkov.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Volkov.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: Volkov.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Volkov %DVIPSParameters: dpi=600, 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Fm(r)1670 3716 y Fr(\()p Fo(\017)1747 3731 y Fm(r)1786 3716 y Fl(p)p Fr(\))p Fo(;)17 b(u)1986 3731 y Fm(s)2022 3716 y Fr(\()p Fo(\017)2099 3731 y Fm(s)2136 3716 y Fl(p)p Fr(\))p Fn(i)2275 3745 y Fe(C)2317 3727 y Fg(4)2389 3716 y Fr(=)28 b Fo(\016)2536 3731 y Fm(r)n(;s)2622 3716 y Fo(:)498 b Fr(\(18\))229 3944 y(W)-8 b(e)39 b(remark)e(that)h(equation) g(\(9\))f(is)g(actually)g(not)h(the)g(original)d(form)h(that)i(w)m(as) 83 4064 y(published)49 b(b)m(y)i(V)-8 b(olk)m(o)m(v,)1006 4028 y Fx(1)1099 4064 y Fr(who)49 b(considered)i(only)d(the)i(sup)s (erp)s(osition)f(of)f(at)h(most)83 4184 y(coun)m(tably)33 b(man)m(y)f(unifrequen)m(t)h(electromagen)m(tic)e(w)m(a)m(v)m(es.)46 b(In)33 b(his)f(form)m(ulas,)f(in\014nite)83 4305 y(series)e(o)m(v)m (er)g(the)g(amplitudes)e(and)h(phases)i(of)d(the)i(external)f(p)s(oten) m(tial)f(o)s(ccur.)42 b(Bro)m(wn)83 4425 y(and)e(Kibble)563 4389 y Fx(2)642 4425 y Fr(and)g(Nikisho)m(v)g(and)g(Ritus)1675 4389 y Fx(7)1754 4425 y Fr(seem)h(to)f(b)s(e)g(the)h(\014rst)g(to)e(ha) m(v)m(e)j(written)83 4545 y(do)m(wn)37 b(the)g(solutions)f(in)f(a)h (form)g(that)g(in)m(v)m(olv)m(es)h(a)f(more)g(or)g(less)h(arbitrary)e (form)g(of)83 4666 y(the)e(electromagnetic)d(p)s(oten)m(tial)h(whic)m (h)h(in)m(v)m(olv)m(es)h(an)g(in)m(tegral)d(in)h(the)i(phase)g(factor.) 83 4786 y(A)d(recen)m(t)h(deriv)-5 b(ation)28 b(of)h(the)i(V)-8 b(olk)m(o)m(v)30 b(solutions)e(can)i(b)s(e)g(found)g(in,)2609 4750 y Fx(18)2714 4786 y Fr(where)h(also)d(the)83 4907 y(cases)41 b Fo(r)h Fr(=)d(3)p Fo(;)17 b Fr(4)38 b(are)i(treated.)64 b(In)39 b(addition,)h(w)m(e)g(men)m(tion)e(that)i(the)f(authors)h (cited)83 5027 y(ab)s(o)m(v)m(e)35 b(use)h Fn(\0001)f Fr(instead)f(of)h(0)f(as)h(the)g(lo)m(w)m(er)g(in)m(tegration)e(limit)e (in)j(the)h(phase)g(\(12\).)83 5147 y(T)-8 b(aking)35 b(0)g(\(or)g(an)m(y)g(other)h(\014nite)f(n)m(um)m(b)s(er\))g(do)s(es)h (not)f(require)h(the)f(v)m(ector)i(p)s(oten)m(tial)83 5268 y(to)j(deca)m(y)h(for)f(large)f(v)-5 b(alues)39 b Fn(j)p Fo(\030)5 b Fn(j)39 b Fr(of)h(the)g(argumen)m(t)g Fo(\030)k Fr(in)c(order)g(for)f(the)i(in)m(tegral)d(to)83 5388 y(con)m(v)m(erge.)1678 5637 y(5)p eop %%Page: 6 6 6 5 bop 578 573 a Fr(F)-8 b(or)32 b(the)h(sak)m(e)h(of)e(simplicit)m(y) -8 b(,)30 b(w)m(e)j(set)h Fl(n)28 b Fr(=)f Fl(e)2248 588 y Fm(z)2288 573 y Fr(,)33 b(so)f(that)1713 804 y Fo(t)23 b Fn(\000)f Fl(n)h Fn(\001)e Fl(r)28 b Fr(=)f Fo(t)c Fn(\000)f Fo(z)1113 b Fr(\(19\))432 1035 y(and)1491 1167 y Fo(E)6 b Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fl(n)f Fn(\001)g Fl(p)28 b Fr(=)g Fo(E)6 b Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(p)2544 1182 y Fm(z)2583 1167 y Fo(:)886 b Fr(\(20\))432 1354 y(The)33 b(spatial)e(part)h(of)h(the)g(v)m(ector)g (p)s(oten)m(tial)e(is)h(then)h(of)f(the)h(form)910 1585 y Fl(A)p Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(=)1352 1560 y(~)1326 1585 y Fo(A)1399 1600 y Fm(x)1443 1585 y Fr(\()p Fo(t)c Fn(\000)f Fo(z)t Fr(\))p Fl(e)1776 1600 y Fm(x)1843 1585 y Fr(+)1967 1560 y(~)1941 1585 y Fo(A)2014 1600 y Fm(y)2056 1585 y Fr(\()p Fo(t)g Fn(\000)h Fo(z)t Fr(\))p Fl(e)2389 1600 y Fm(y)2459 1585 y Fn(\021)2578 1561 y Fl(~)2564 1585 y(A)o Fr(\()p Fo(t)g Fn(\000)f Fo(z)t Fr(\))29 b Fn(2)f Fj(R)3119 1544 y Fx(2)3164 1585 y Fo(:)305 b Fr(\(21\))432 1816 y(Here)33 b(as)g(in)f(the)h(follo)m (wing,)d(w)m(e)j(mak)m(e)g(the)g(iden)m(ti\014cation)1616 2048 y Fj(R)1682 2006 y Fx(2)1755 2020 y Fn(\030)1756 2052 y Fr(=)1860 2048 y Fj(R)1926 2006 y Fx(2)1993 2048 y Fn(\002)23 b(f)p Fr(0)p Fn(g)k(\032)h Fj(R)2440 2006 y Fx(3)3496 2048 y Fr(\(22\))432 2279 y(in)k(order)g(to)g(suppress)j (an)e(o)m(v)m(erduly)g(complex)f(notation.)43 b(W)-8 b(e)33 b(also)e(set)1643 2510 y Fl(p)1705 2525 y Fy(?)1792 2510 y Fr(:=)c(\()p Fo(p)2009 2525 y Fm(x)2053 2510 y Fo(;)17 b(p)2146 2525 y Fm(y)2187 2510 y Fr(\))28 b Fn(2)g Fj(R)2413 2469 y Fx(2)3496 2510 y Fr(\(23\))432 2741 y(and)1676 2873 y Fl(r)1722 2888 y Fy(?)1808 2873 y Fr(:=)g(\()p Fo(x;)17 b(y)t Fr(\))27 b Fn(2)h Fj(R)2353 2831 y Fx(2)2398 2873 y Fo(;)1071 b Fr(\(24\))432 3060 y(so)32 b(that)1482 3191 y Fl(p)22 b Fn(\001)g Fl(A)o Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))28 b(=)f Fl(p)2094 3206 y Fy(?)2176 3191 y Fn(\001)2240 3167 y Fl(~)2225 3191 y(A)p Fr(\()p Fo(t)22 b Fn(\000)h Fo(z)t Fr(\))p Fo(:)877 b Fr(\(25\))432 3379 y(In)37 b(addition,)g(w)m(e)h(will)d(assume)i(throughout)g(that)g(the)h (functions)3008 3354 y(~)2982 3379 y Fo(A)3055 3394 y Fm(x)3099 3379 y Fo(;)3168 3354 y Fr(~)3143 3379 y Fo(A)3216 3394 y Fm(y)3293 3379 y Fr(:)d Fj(R)47 b Fn(!)35 b Fj(R)432 3499 y Fr(are)d Fn(C)646 3514 y Fm(b)681 3499 y Fr(,)h(i.)e(e.)44 b(con)m(tin)m(uous)34 b(and)e(b)s(ounded,)i(and)e(set)1275 3772 y Fn(j)o(j)p Fl(A)p Fn(j)o(j)c Fr(:=)f(sup)1637 3854 y Fm(\030)s Fy(2)p Fe(R)1792 3646 y Fk(q)p 1891 3646 678 4 v 1917 3747 a Fr(~)1891 3772 y Fo(A)1964 3787 y Fm(x)2008 3772 y Fr(\()p Fo(\030)5 b Fr(\))2132 3744 y Fx(2)2193 3772 y Fr(+)2317 3747 y(~)2291 3772 y Fo(A)2364 3787 y Fm(y)2406 3772 y Fr(\()p Fo(\030)g Fr(\))2530 3744 y Fx(2)2596 3772 y Fo(<)28 b Fn(1)p Fo(:)669 b Fr(\(26\))432 4158 y Fq(I)t(I)t(I.)55 b(Symmetries)d(of)i(the)e(V)-13 b(olk)l(o)l(v)53 b(solutions)432 4385 y Fr(The)39 b(construction)g(of)f (the)i(V)-8 b(olk)m(o)m(v)38 b(solutions)g(\(9\))g(relies)g(hea)m(vily) h(on)f(certain)h(sym-)432 4505 y(metries)i(of)h(the)g(Dirac)f(equation) g(\(1\):)62 b(The)43 b(three)g(di\013eren)m(tial)d(op)s(erators)i Fo(@)3434 4520 y Fm(x)3478 4505 y Fr(,)j Fo(@)3601 4520 y Fm(y)3642 4505 y Fr(,)432 4626 y(and)34 b Fo(@)674 4641 y Fm(t)727 4626 y Fr(+)23 b Fo(@)877 4641 y Fm(z)951 4626 y Fr(comm)m(ute)33 b(formally)e(with)1981 4545 y Fk(\000)2027 4626 y Fo(i)2070 4616 y(=)2060 4626 y(@)d Fn(\000)23 b Fo(e)2303 4617 y(=)2284 4626 y(A)f Fn(\000)h Fo(m)2564 4545 y Fk(\001)2610 4626 y Fr(,)34 b(whic)m(h)g(has)g (already)g(b)s(een)432 4746 y(noticed)39 b(b)m(y)i(Nikisho)m(v)e(and)h (Ritus.)1785 4710 y Fx(7)1889 4746 y Fr(These)h(translational)c (symmetries)i(allo)m(w)f(the)432 4866 y(separation)32 b(of)g(v)-5 b(ariables)31 b(in)h(the)h(equation,)f(whic)m(h)h(is)g (inherited)e(b)m(y)j(the)f(relations)1493 5098 y Fn(\000)p Fo(i@)1654 5113 y Fm(x)1700 5098 y Fo( )1763 5113 y Fi(p)p Fm(;r)1949 5098 y Fr(=)83 b Fo(\017)2147 5113 y Fm(r)2185 5098 y Fo(p)2234 5113 y Fm(x)2278 5098 y Fo( )2341 5113 y Fi(p)p Fm(;r)2444 5098 y Fo(;)1025 b Fr(\(27\))1496 5243 y Fn(\000)p Fo(i@)1657 5258 y Fm(y)1700 5243 y Fo( )1763 5258 y Fi(p)p Fm(;r)1949 5243 y Fr(=)83 b Fo(\017)2147 5258 y Fm(r)2185 5243 y Fo(p)2234 5258 y Fm(y)2275 5243 y Fo( )2338 5258 y Fi(p)p Fm(;r)2442 5243 y Fo(;)1027 b Fr(\(28\))1187 5388 y Fn(\000)p Fo(i)17 b Fr(\()q Fo(@)1404 5403 y Fm(t)1456 5388 y Fr(+)22 b Fo(@)1605 5403 y Fm(z)1645 5388 y Fr(\))17 b Fo( )1763 5403 y Fi(p)p Fm(;r)1949 5388 y Fr(=)83 b Fo(\017)2147 5403 y Fm(r)2202 5388 y Fr(\()p Fo(p)2289 5403 y Fm(z)2350 5388 y Fn(\000)23 b Fo(E)6 b Fr(\()p Fl(p)p Fr(\)\))17 b Fo( )2784 5403 y Fi(p)p Fm(;r)2887 5388 y Fo(:)582 b Fr(\(29\))2026 5637 y(6)p eop %%Page: 7 7 7 6 bop 83 573 a Fr(\(F)-8 b(or)25 b(more)g(symmetries)g(of)g(the)h (Dirac)f(equation)g(\(1\))h(and)f(a)h(detailed)e(group)i(theoret-)83 693 y(ical)31 b(analysis,)g(see)j(the)f(w)m(ork)g(of)f(Janner)h(and)f (Janssen.)2193 657 y Fx(24)2270 693 y Fr(\))43 b(Using)32 b(the)h(Hamiltonian)83 814 y(form)e(of)h(\(1\),)h(i.)16 b(e.)1212 934 y Fo(i@)1296 949 y Fm(t)1326 934 y Fo( )t Fr(\()p Fl(r)p Fo(;)h(t)p Fr(\))27 b(=)h Fo(H)8 b Fr(\()p Fo(t)p Fr(\))p Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))954 b(\(30\))83 1089 y(with)32 b(the)h(formally)d(self-adjoin)m(t)h (Hamiltonian)e(op)s(erator)1077 1264 y Fo(H)8 b Fr(\()p Fo(t)p Fr(\))27 b(=)h Ff(\013)16 b Fr(\()p Fn(\000)p Fo(i)p Fd(r)23 b Fn(\000)g Fo(e)p Fl(A)p Fr(\))f(+)g Fo(\014)6 b(m;)820 b Fr(\(31\))83 1439 y(w)m(e)34 b(easily)d(deriv)m(e) j(from)d(\(29\))h(that)g(for)g(ev)m(ery)j Fo(t)p Fr(,)596 1614 y(\()p Fo(H)8 b Fr(\()p Fo(t)p Fr(\))22 b(+)g Fo(i@)1038 1629 y Fm(z)1079 1614 y Fr(\))16 b Fo( )1196 1629 y Fi(p)p Fm(;r)1327 1614 y Fr(=)28 b Fo(\017)1470 1629 y Fm(r)1524 1614 y Fr(\()p Fo(E)6 b Fr(\()p Fl(p)p Fr(\))23 b Fn(\000)f Fo(p)1949 1629 y Fm(z)1989 1614 y Fr(\))17 b Fo( )2107 1629 y Fi(p)p Fm(;r)2238 1614 y Fn(\021)28 b Fo(\017)2382 1629 y Fm(r)2420 1614 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p Fo( )2677 1629 y Fi(p)p Fm(;r)2780 1614 y Fo(;)340 b Fr(\(32\))83 1789 y(where)34 b(w)m(e)f(ha)m(v)m(e)h(in)m(tro)s(duced) 787 1988 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))28 b(:=)g Fo(E)6 b Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(p)1527 2003 y Fm(z)1594 1988 y Fr(=)1698 1859 y Fk(q)p 1797 1859 402 4 v 1797 1988 a Fo(m)1882 1959 y Fx(2)1944 1988 y Fr(+)f Fn(j)p Fl(p)p Fn(j)2160 1939 y Fx(2)2221 1988 y Fn(\000)h Fo(p)2370 2003 y Fm(z)2437 1988 y Fo(>)28 b Fr(0)p Fo(:)530 b Fr(\(33\))83 2163 y(The)30 b(V)-8 b(olk)m(o)m(v)29 b(solutions)f(are)h(th)m(us)h(sim)m(ultaneous)e (eigenfunctions)h(of)f(the)i(three)f(op)s(er-)83 2283 y(ators)34 b Fn(\000)p Fo(i@)490 2298 y Fm(x)535 2283 y Fr(,)h Fn(\000)p Fo(i@)758 2298 y Fm(y)801 2283 y Fr(,)g(and)f Fo(H)8 b Fr(\()p Fo(t)p Fr(\))23 b(+)g Fo(i@)1460 2298 y Fm(z)1535 2283 y Fr(with)34 b(corresp)s(onding)h(eigen)m(v)-5 b(alues)34 b Fo(\017)2933 2298 y Fm(r)2971 2283 y Fo(p)3020 2298 y Fm(x)3064 2283 y Fr(,)h Fo(\017)3165 2298 y Fm(r)3203 2283 y Fo(p)3252 2298 y Fm(y)3294 2283 y Fr(,)83 2403 y(and)h Fo(\017)315 2418 y Fm(r)353 2403 y Fo(\024)p Fr(\()p Fl(p)p Fr(\),)h(whic)m(h)g(do)f(not)g(dep)s(end)h(on)e(the)i(c) m(hosen)g(time)e Fo(t)p Fr(.)53 b(It)36 b(will)e(b)s(e)i(useful)g(to)83 2524 y(de\014ne)742 2644 y Fl(^)-64 b Ff(\034)41 b Fr(:)28 b Fj(R)948 2603 y Fx(3)1021 2644 y Fn(!)f Fj(R)1214 2603 y Fx(2)1282 2644 y Fn(\002)c Fj(R)1448 2603 y Fx(+)1540 2644 y Fo(;)45 b Fl(p)28 b Fn(7!)35 b Fl(^)-64 b Ff(\034)13 b Fr(\()p Fl(p)p Fr(\))28 b(:=)g(\()p Fl(p)2291 2659 y Fy(?)2350 2644 y Fo(;)17 b(\024)p Fr(\()p Fl(p)p Fr(\)\))f Fo(:)478 b Fr(\(34\))83 2799 y(Note)33 b(that)f(the)h(in)m(v)m(erse)h (is)e(giv)m(en)h(b)m(y)224 3056 y Ff(\034)41 b Fr(:=)36 b Fl(^)-64 b Ff(\034)513 3013 y Fy(\000)p Fx(1)635 3056 y Fr(:)28 b Fj(R)756 3014 y Fx(2)823 3056 y Fn(\002)23 b Fj(R)989 3014 y Fx(+)1082 3056 y Fn(!)k Fj(R)1275 3014 y Fx(3)1348 3056 y Fo(;)44 b Ff(\034)14 b Fr(\()p Fl(p)1585 3071 y Fy(?)1644 3056 y Fo(;)j(\025)p Fr(\))27 b(:=)1941 2885 y Fk( )2020 3056 y Fl(p)2082 3071 y Fy(?)2141 3056 y Fo(;)2195 2988 y(m)2280 2952 y Fx(2)2342 2988 y Fr(+)22 b Fn(j)o Fl(p)2529 3003 y Fy(?)2589 2988 y Fn(j)2616 2940 y Fx(2)2678 2988 y Fn(\000)g Fo(\025)2834 2952 y Fx(2)p 2195 3033 680 4 v 2481 3124 a Fr(2)p Fo(\025)2884 2885 y Fk(!)2979 3056 y Fo(:)141 b Fr(\(35\))229 3318 y(The)34 b(matrix)d(part)i(\(11\))f(and)h(the)h(phase)f(\(12\))g(of)f (the)h(V)-8 b(olk)m(o)m(v)33 b(solutions)f(ma)m(y)h(b)s(e)83 3438 y(rewritten)g(in)f(a)g(simple)f(form)g(in)h(terms)h(of)f(the)h (eigen)m(v)-5 b(alues:)988 3695 y Fl(M)1094 3654 y Fx(\()p Fm(\017)1150 3662 y Fh(r)1184 3654 y Fx(\))1216 3695 y Fr(\()p Fl(p)p Fo(;)17 b(\030)5 b Fr(\))27 b(=)h Fl(1)22 b Fr(+)1773 3627 y Fo(e)1823 3630 y(=)1818 3627 y(n)1889 3586 y Fr(~)1896 3618 y Fo(=)1876 3627 y(A)q Fr(\()p Fo(\030)5 b Fr(\))p 1763 3672 321 4 v 1763 3763 a(2)p Fo(\017)1851 3778 y Fm(r)1889 3763 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))2121 3695 y Fn(2)28 b Fj(C)2281 3654 y Fx(4)p Fy(\002)p Fx(4)3147 3695 y Fr(\(36\))83 3920 y(and)175 4133 y Fo(S)241 4092 y Fx(\()p Fm(\017)297 4100 y Fh(r)332 4092 y Fx(\))363 4133 y Fr(\()p Fl(p)p Fo(;)17 b Fl(r)p Fo(;)g(t)p Fr(\))83 b(=)g Fo(\017)951 4148 y Fm(r)989 4133 y Fl(p)1051 4148 y Fy(?)1133 4133 y Fn(\001)21 b Fl(r)1228 4148 y Fy(?)1309 4133 y Fn(\000)i Fo(\017)1448 4148 y Fm(r)1486 4133 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p Fo(t)g Fn(\000)g Fo(\017)1877 4148 y Fm(r)1915 4133 y Fo(p)1964 4148 y Fm(z)2004 4133 y Fr(\()p Fo(t)f Fn(\000)h Fo(z)t Fr(\))g Fn(\000)f Fo(\017)2447 4148 y Fm(r)2502 3997 y Fk(Z)2602 4024 y Fm(t)p Fy(\000)p Fm(z)2557 4223 y Fx(0)2738 4133 y Fo(J)2801 4092 y Fx(\()p Fm(\017)2857 4100 y Fh(r)2892 4092 y Fx(\))2792 4158 y Fi(p)2924 4133 y Fr(\()p Fo(\030)5 b Fr(\))17 b Fo(d\030)753 4430 y Fr(=)83 b Fo(\017)951 4445 y Fm(r)989 4430 y Fl(p)1051 4445 y Fy(?)1133 4430 y Fn(\001)21 b Fl(r)1228 4445 y Fy(?)1309 4430 y Fn(\000)i Fo(\017)1448 4445 y Fm(r)1486 4430 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p Fo(t)g Fn(\000)g Fo(\017)1877 4445 y Fm(r)1915 4260 y Fk(")17 b( )2069 4430 y Fo(p)2118 4445 y Fm(z)2180 4430 y Fn(\000)2457 4363 y(j)o Fl(p)2546 4378 y Fy(?)2606 4363 y Fn(j)2633 4315 y Fx(2)p 2289 4408 552 4 v 2289 4499 a Fr(2\()p Fo(E)6 b Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(p)2763 4514 y Fm(z)2803 4499 y Fr(\))2850 4260 y Fk(!)2946 4430 y Fr(\()p Fo(t)f Fn(\000)h Fo(z)t Fr(\))912 4763 y(+)1249 4695 y(1)p 998 4740 V 998 4831 a(2\()p Fo(E)6 b Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)g Fo(p)1471 4846 y Fm(z)1511 4831 y Fr(\))1576 4627 y Fk(Z)1675 4653 y Fm(t)p Fy(\000)p Fm(z)1631 4853 y Fx(0)1812 4648 y Fk(\014)1812 4708 y(\014)1812 4768 y(\014)1845 4763 y Fo(\017)1884 4778 y Fm(r)1923 4763 y Fl(p)1985 4778 y Fy(?)2066 4763 y Fn(\000)h Fo(e)2225 4739 y Fl(~)2211 4763 y(A)o Fr(\()p Fo(\030)5 b Fr(\))2419 4648 y Fk(\014)2419 4708 y(\014)2419 4768 y(\014)2452 4674 y Fx(2)2525 4763 y Fo(d\030)2624 4592 y Fk(#)753 5067 y Fr(=)83 b Fo(\017)951 5082 y Fm(r)989 5067 y Fl(p)1051 5082 y Fy(?)1133 5067 y Fn(\001)21 b Fl(r)1228 5082 y Fy(?)1309 5067 y Fn(\000)i Fo(\017)1448 5082 y Fm(r)1486 5067 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p Fo(t)g Fn(\000)1848 4999 y Fr(1)p 1848 5044 49 4 v 1848 5135 a(2)1923 4926 y Fk(\022)2080 4999 y Fo(m)2165 4963 y Fx(2)p 2006 5044 272 4 v 2006 5135 a Fo(\017)2045 5150 y Fm(r)2084 5135 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))2310 5067 y Fn(\000)g Fo(\017)2449 5082 y Fm(r)2487 5067 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))2681 4926 y Fk(\023)2771 5067 y Fr(\()p Fo(t)g Fn(\000)f Fo(z)t Fr(\))912 5345 y Fn(\000)1135 5277 y Fr(1)p 999 5322 321 4 v 999 5413 a(2)p Fo(\017)1087 5428 y Fm(r)1125 5413 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))1346 5209 y Fk(Z)1446 5235 y Fm(t)p Fy(\000)p Fm(z)1402 5435 y Fx(0)1583 5230 y Fk(\014)1583 5290 y(\014)1583 5350 y(\014)1616 5345 y Fo(\017)1655 5360 y Fm(r)1693 5345 y Fl(p)1755 5360 y Fy(?)1837 5345 y Fn(\000)g Fo(e)1995 5321 y Fl(~)1981 5345 y(A)p Fr(\()p Fo(\030)5 b Fr(\))2190 5230 y Fk(\014)2190 5290 y(\014)2190 5350 y(\014)2223 5256 y Fx(2)2295 5345 y Fo(d\030)g(:)726 b Fr(\(37\))1678 5637 y(7)p eop %%Page: 8 8 8 7 bop 432 573 a Fr(A)46 b(similar)e(form)h(of)h(the)h(phase)h Fo(S)1789 537 y Fx(\()p Fm(\017)1845 545 y Fh(r)1879 537 y Fx(\))1957 573 y Fr(has)f(already)f(b)s(een)i(found)f(b)m(y)g (Gol'dman.)3632 537 y Fx(3)432 693 y Fr(De\014ning)818 1074 y Fo(\031)873 1089 y Fy(?)932 1074 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)1124 1089 y Fy(?)1182 1074 y Fr(\))28 b(:=)1379 840 y Fk(8)1379 930 y(>)1379 960 y(<)1379 1139 y(>)1379 1169 y(:)1509 861 y(\022)1592 934 y Fr(1)p 1592 979 49 4 v 1593 1070 a Fo(\030)1667 866 y Fk(Z)1767 893 y Fm(\030)1723 1092 y Fx(0)1821 887 y Fk(\014)1821 947 y(\014)1821 1007 y(\014)1854 1002 y Fl(p)1916 1017 y Fy(?)1998 1002 y Fn(\000)23 b Fo(e)2157 978 y Fl(~)2143 1002 y(A)o Fr(\()p Fo(\030)2313 961 y Fy(0)2336 1002 y Fr(\))2374 887 y Fk(\014)2374 947 y(\014)2374 1007 y(\014)2407 914 y Fx(2)2480 1002 y Fo(d\030)2579 961 y Fy(0)2601 861 y Fk(\023)2675 873 y Fx(1)p Fm(=)p Fx(2)2868 1002 y Fo(;)83 b(\030)32 b Fn(6)p Fr(=)27 b(0)p Fo(;)1865 1096 y Fk(\014)1865 1156 y(\014)1865 1216 y(\014)1898 1211 y Fl(p)1960 1226 y Fy(?)2041 1211 y Fn(\000)c Fo(e)2200 1187 y Fl(~)2186 1211 y(A)p Fr(\(0\))2396 1096 y Fk(\014)2396 1156 y(\014)2396 1216 y(\014)2868 1211 y Fo(;)83 b(\030)32 b Fr(=)27 b(0)p Fo(;)3496 1074 y Fr(\(38\))469 1535 y Ff(\027)533 1550 y Fm(\030)599 1535 y Fr(:)h Fj(R)720 1494 y Fx(2)787 1535 y Fn(\002)23 b Fr(\()p Fj(R)33 b Fn(n)22 b(f)p Fr(0)p Fn(g)p Fr(\))27 b Fn(!)g Fj(R)1498 1494 y Fx(3)1571 1535 y Fo(;)45 b Ff(\027)1707 1550 y Fm(\030)1745 1535 y Fr(\()p Fl(p)1845 1550 y Fy(?)1904 1535 y Fo(;)17 b(\025)p Fr(\))27 b(:=)2201 1394 y Fk(\022)2274 1535 y Fl(p)2336 1550 y Fy(?)2395 1535 y Fo(;)2449 1467 y Fr(1)p 2449 1512 V 2449 1603 a(2)2524 1394 y Fk(\024)2587 1467 y Fo(m)2672 1431 y Fx(2)2734 1467 y Fr(+)22 b Fo(\031)2887 1482 y Fy(?)2946 1467 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)3138 1482 y Fy(?)3197 1467 y Fr(\))3235 1431 y Fx(2)p 2587 1512 688 4 v 2902 1603 a Fo(\025)3306 1535 y Fn(\000)23 b Fo(\025)3463 1394 y Fk(\025)o(\023)3605 1535 y Fo(;)3496 1712 y Fr(\(39\))890 1948 y Fl(q)949 1897 y Fx(\()p Fy(\006)p Fx(\))949 1976 y Fm(\030)1063 1948 y Fr(\()p Fl(p)p Fr(\))84 b(:=)f Ff(\027)1535 1963 y Fm(\030)1589 1948 y Fr(\()p Fn(\006)8 b Fl(^)-64 b Ff(\034)14 b Fr(\()p Fl(p)p Fr(\)\))1298 2169 y(=)97 b Fn(\006)1565 2028 y Fk(\022)1638 2169 y Fl(p)1700 2184 y Fy(?)1759 2169 y Fo(;)1813 2101 y Fr(1)p 1813 2146 49 4 v 1813 2237 a(2)1888 2028 y Fk(\024)1951 2101 y Fo(m)2036 2065 y Fx(2)2098 2101 y Fr(+)22 b Fo(\031)2251 2116 y Fy(?)2310 2101 y Fr(\()p Fo(\030)5 b(;)17 b Fn(\006)p Fl(p)2579 2116 y Fy(?)2638 2101 y Fr(\))2676 2065 y Fx(2)p 1951 2146 765 4 v 2236 2237 a Fo(\024)p Fr(\()p Fl(p)p Fr(\))2747 2169 y Fn(\000)23 b Fo(\024)p Fr(\()p Fl(p)p Fr(\))3041 2028 y Fk(\025\023)3184 2169 y Fo(;)285 b Fr(\(40\))432 2443 y(and)1622 2565 y Fl(s)1666 2580 y Fm(t)1696 2565 y Fr(\()p Fl(r)p Fr(\))27 b(:=)h(\()p Fo(x;)17 b(y)t(;)g(z)26 b Fn(\000)c Fo(t)p Fr(\))p Fo(;)1017 b Fr(\(41\))432 2742 y(w)m(e)33 b(ma)m(y)g(write)f(this)g(as)1216 2965 y Fo(S)1282 2924 y Fx(\()p Fm(\017)1338 2932 y Fh(r)1373 2924 y Fx(\))1404 2965 y Fr(\()p Fl(p)p Fo(;)17 b Fl(r)p Fo(;)g(t)p Fr(\))28 b(=)f Fl(q)1901 2914 y Fx(\()p Fm(\017)1957 2922 y Fh(r)1992 2914 y Fx(\))1901 2987 y Fm(t)p Fy(\000)p Fm(z)2024 2965 y Fr(\()p Fl(p)p Fr(\))22 b Fn(\001)g Fl(s)2278 2980 y Fm(t)2307 2965 y Fr(\()p Fl(r)p Fr(\))g Fn(\000)h Fo(\017)2590 2980 y Fm(r)2628 2965 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p Fo(t:)612 b Fr(\(42\))432 3187 y(Note)32 b(that)1366 3310 y(0)27 b Fn(\024)h Fo(\031)1602 3325 y Fy(?)1662 3310 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)1854 3325 y Fy(?)1912 3310 y Fr(\))28 b Fn(\024)g(j)o Fl(p)2172 3325 y Fy(?)2232 3310 y Fn(j)21 b Fr(+)i Fn(j)p Fo(e)p Fn(j)17 b(j)n(j)p Fl(A)p Fn(j)o(j)g Fo(:)760 b Fr(\(43\))432 3487 y(Moreo)m(v)m(er,)32 b(for)1047 3463 y Fl(~)1033 3487 y(A)27 b Fn(\021)h Fr(0,)j Fo(\031)1412 3502 y Fy(?)1471 3487 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)1663 3502 y Fy(?)1722 3487 y Fr(\))27 b(=)h Fn(j)p Fl(p)1981 3502 y Fy(?)2040 3487 y Fn(j)p Fr(,)j(so)g(that)f Fo(\031)2508 3502 y Fy(?)2568 3487 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)2760 3502 y Fy(?)2818 3487 y Fr(\))31 b(ma)m(y)f(b)s(e)h(though)m(t)g(of)432 3607 y(as)h(a)h(mo)s(di\014ed)e(\\p)s(erp)s(endicular)h(momen)m(tum".) 432 3945 y Fq(IV.)53 b(Regular)i(w)l(a)l(v)l(e)d(pac)l(k)l(ets)432 4166 y Fr(In)39 b(this)f(section,)j(w)m(e)f(shall)d(construct)j (square-in)m(tegrable)f(w)m(a)m(v)m(e)i(pac)m(k)m(ets)f(from)e(the)432 4286 y(V)-8 b(olk)m(o)m(v)32 b(solutions.)43 b(These)34 b(w)m(a)m(v)m(e)h(pac)m(k)m(ets)f(will)d(b)s(e)h(of)h(the)g(form)1135 4593 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(=)1666 4525 y(1)p 1544 4570 294 4 v 1544 4661 a(\(2)p Fo(\031)t Fr(\))1728 4632 y Fx(3)p Fm(=)p Fx(2)1918 4468 y(4)1864 4498 y Fk(X)1874 4707 y Fm(r)r Fx(=1)2024 4457 y Fk(Z)2080 4683 y Fe(R)2128 4664 y Fg(3)2183 4593 y Fo(\036)2241 4608 y Fm(r)2279 4593 y Fr(\()p Fl(p)p Fr(\))p Fo( )2480 4608 y Fi(p)p Fm(;r)2583 4593 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)2852 4552 y Fx(3)2890 4593 y Fo(p:)530 b Fr(\(44\))432 4907 y(This)29 b(in)m(tegral)f(represen)m(tation)i(w)m (as)h(used,)g(for)e(example,)h(b)m(y)g(San)g(Rom\023)-49 b(an)28 b Fp(et)k(al.)3489 4870 y Fx(18{)q(21)432 5027 y Fr(in)26 b(their)h(calculations)f(of)g(the)i(time)e(ev)m(olution)h (of)g(w)m(a)m(v)m(e)i(pac)m(k)m(ets.)44 b(It)27 b(will)e(b)s(e)j(our)f (task)432 5147 y(to)j(mak)m(e)g(sense)j(of)k(\(44\))o(,)31 b(i.)16 b(e.)43 b(w)m(e)31 b(will)d(ensure)k(the)f(existence)h(of)e (this)g(in)m(tegral)f(for)h(as)432 5268 y(man)m(y)i(functions)g Fo(\036)1175 5283 y Fm(r)1245 5268 y Fr(as)g(p)s(ossible)f(and)h(to)g (sho)m(w)h(that)f(the)h(resulting)e(w)m(a)m(v)m(e)j(pac)m(k)m(et)g Fo( )432 5388 y Fr(ful\014lls)c(the)j(Dirac)e(equation)i(\(1\))o(.)2026 5637 y(8)p eop %%Page: 9 9 9 8 bop 229 573 a Fr(In)41 b(this)g(section,)i(w)m(e)e(will)e(start)h (with)h(momen)m(tum)e(distributions)g Fo(\036)2893 588 y Fm(r)2971 573 y Fr(that)i(are)83 693 y(in)f Fn(C)263 657 y Fy(1)257 718 y Fx(0)379 693 y Fr(and)h(sho)m(w)h(that)f(the)g(w)m (a)m(v)m(e)i(pac)m(k)m(ets)g(de\014ned)f(b)m(y)g(these)g(deca)m(y)h (faster)e(than)83 814 y(an)m(y)d(p)s(o)m(w)m(er)g(of)f Fn(j)p Fl(r)p Fn(j)g Fr(for)g Fn(j)o Fl(r)p Fn(j)f(!)f(1)p Fr(.)57 b(In)38 b(order)g(to)f(con)m(trol)f(the)i Fl(p)g Fr(dep)s(endence)i(of)c(the)83 948 y(phase)26 b(\(42\),)g(w)m(e)g(need) g(to)f(determine)f(the)i(in)m(v)m(erses)h(of)d(the)i(mappings)e Fl(q)2759 897 y Fx(\()p Fy(\006)p Fx(\))2759 975 y Fm(\030)2898 948 y Fr(from)f(\(40\):)424 1202 y Fl(~)-57 b(q)482 1151 y Fx(\()p Fy(\006)p Fx(\))482 1230 y Fm(\030)596 1202 y Fr(\()s Fl(~)e(p)p Fr(\))83 b(:=)1003 1091 y Fk(\020)1062 1202 y Fl(q)1121 1151 y Fx(\()p Fy(\006)p Fx(\))1121 1230 y Fm(\030)1236 1091 y Fk(\021)1295 1114 y Fy(\000)p Fx(1)1406 1202 y Fr(\()s Fl(~)-59 b(p)p Fr(\))831 1385 y(=)96 b Ff(\034)1085 1304 y Fk(\000)1131 1385 y Fn(\006)p Ff(\027)1272 1343 y Fy(\000)p Fx(1)1272 1412 y Fm(\030)1367 1385 y Fr(\()s Fl(~)-59 b(p)p Fr(\))1505 1304 y Fk(\001)831 1603 y Fr(=)96 b Ff(\034)1085 1462 y Fk(\022)1158 1603 y Fn(\006)1252 1462 y Fk(\022)1326 1603 y Fl(p)1388 1618 y Fy(?)1447 1603 y Fo(;)1491 1475 y Fk(q)p 1590 1475 991 4 v 1590 1603 a Fo(m)1675 1574 y Fx(2)1737 1603 y Fr(+)22 b Fo(\031)1890 1618 y Fy(?)1966 1603 y Fr(\()p Fo(\030)5 b(;)17 b Fn(\006)s Fl(~)-59 b(p)2235 1618 y Fy(?)2294 1603 y Fr(\))2332 1554 y Fx(2)2394 1603 y Fr(+)30 b(~)-57 b Fo(p)2541 1574 y Fx(2)2541 1628 y Fm(z)2602 1603 y Fn(\007)31 b Fr(~)-57 b Fo(p)2751 1618 y Fm(z)2791 1462 y Fk(\023\023)2954 1603 y Fo(;)166 b Fr(\(45\))83 1849 y(whic)m(h)33 b(are)g Fn(C)583 1813 y Fy(1)658 1849 y Fr(.)43 b(W)-8 b(e)33 b(claim:)83 2031 y Fl(Lemma)40 b(1.)i Fp(F)-7 b(or)36 b(every)h(c)-5 b(omp)g(act)36 b(set)h Fo(K)i Fn(\032)32 b Fj(R)1885 1995 y Fx(3)1967 2031 y Fp(and)k(every)h Fo(n)2476 2046 y Fm(x)2520 2031 y Fo(;)17 b(n)2622 2046 y Fm(y)2663 2031 y Fo(;)g(n)2765 2046 y Fm(z)2836 2031 y Fn(2)32 b Fj(N)3000 2046 y Fx(0)3046 2031 y Fp(,)37 b(ther)-5 b(e)83 2151 y(is)35 b(a)f(c)-5 b(onstant)35 b Fo(c)703 2166 y Fm(n)746 2174 y Fh(x)785 2166 y Fm(;n)848 2174 y Fh(y)884 2166 y Fm(;n)947 2174 y Fh(z)987 2151 y Fr(\()p Fo(K)7 b Fr(\))35 b Fp(satisfying)186 2269 y Fk(\014)186 2329 y(\014)186 2389 y(\014)186 2449 y(\014)219 2269 y(\014)219 2329 y(\014)219 2389 y(\014)219 2449 y(\014)252 2273 y(\022)382 2346 y Fo(@)p 335 2391 150 4 v 335 2482 a(@)14 b Fr(~)-58 b Fo(p)440 2497 y Fm(x)495 2273 y Fk(\023)568 2296 y Fm(n)611 2304 y Fh(x)671 2273 y Fk(\022)799 2346 y Fo(@)p 754 2391 148 4 v 754 2482 a(@)14 b Fr(~)-58 b Fo(p)859 2497 y Fm(y)911 2273 y Fk(\023)985 2296 y Fm(n)1028 2304 y Fh(y)1086 2273 y Fk(\022)1213 2346 y Fo(@)p 1169 2391 146 4 v 1169 2482 a(@)14 b Fr(~)-58 b Fo(p)1274 2497 y Fm(z)1325 2273 y Fk(\023)1398 2296 y Fm(n)1441 2304 y Fh(z)1499 2414 y Fl(~)g(q)1556 2363 y Fx(\()p Fy(\006)p Fx(\))1556 2442 y Fm(\030)1671 2414 y Fr(\()s Fl(~)f(p)p Fr(\))1809 2269 y Fk(\014)1809 2329 y(\014)1809 2389 y(\014)1809 2449 y(\014)1842 2269 y(\014)1842 2329 y(\014)1842 2389 y(\014)1842 2449 y(\014)1903 2414 y Fn(\024)28 b Fo(c)2050 2429 y Fm(n)2093 2437 y Fh(x)2132 2429 y Fm(;n)2195 2437 y Fh(y)2232 2429 y Fm(;n)2295 2437 y Fh(z)2334 2414 y Fr(\()p Fo(K)7 b Fr(\))28 b Fo(;)47 b Fl(~)-59 b(p)28 b Fn(2)g Fo(K)c(;)33 b(\030)f Fn(2)c Fj(R)5 b Fo(:)3147 2593 y Fr(\(46\))83 2788 y Fp(Pr)-5 b(o)g(of.)48 b Fr(By)38 b(the)g(c)m(hain)f(rule,)h(one) g(sees)h(that)e(the)g(deriv)-5 b(ativ)m(es)38 b(of)g Fl(~)-58 b(q)2647 2737 y Fx(\()p Fy(\006)p Fx(\))2647 2816 y Fm(\030)2799 2788 y Fr(dep)s(end)38 b(con-)83 2928 y(tin)m(uously)27 b(on)j Fl(~)-59 b(p)28 b Fr(as)f(w)m(ell)f(as)i (on)f Fo(\031)1327 2943 y Fy(?)1386 2928 y Fr(\()p Fo(\030)5 b(;)20 b Fl(~)-59 b(p)1578 2943 y Fy(?)1637 2928 y Fr(\))1675 2892 y Fx(2)1714 2928 y Fr(,)29 b Fo(@)1827 2943 y Fx(~)-41 b Fm(p)1857 2951 y Fh(x)1916 2928 y Fr([)p Fo(\031)1998 2943 y Fy(?)2058 2928 y Fr(\()p Fo(\030)5 b(;)19 b Fl(~)-58 b(p)2250 2943 y Fy(?)2308 2928 y Fr(\)])2373 2880 y Fx(2)2413 2928 y Fr(,)28 b(and)f Fo(@)2709 2943 y Fx(~)-41 b Fm(p)2739 2951 y Fh(y)2797 2928 y Fr([)q Fo(\031)2880 2943 y Fy(?)2939 2928 y Fr(\()p Fo(\030)5 b(;)19 b Fl(~)-58 b(p)3131 2943 y Fy(?)3189 2928 y Fr(\)])3254 2880 y Fx(2)3294 2928 y Fr(.)83 3061 y(\(Note)49 b(that)h(higher-order)e(deriv)-5 b(ativ)m(es)49 b(of)g([)p Fo(\031)1883 3076 y Fy(?)1943 3061 y Fr(\()p Fo(\030)5 b(;)19 b Fl(~)-58 b(p)2135 3076 y Fy(?)2193 3061 y Fr(\)])2258 3013 y Fx(2)2347 3061 y Fr(v)-5 b(anish,)53 b(see)e(the)e(de\014ni-)83 3181 y(tion)43 b(\(38\).\))78 b(The)46 b(last)d(three)i(expressions)h(are)e (b)s(ounded)i(for)d Fo(\030)52 b Fn(2)c Fj(R)56 b Fr(and)47 b Fl(~)-59 b(p)3136 3196 y Fy(?)3240 3181 y Fr(in)83 3302 y(a)38 b(\014xed)i(compact)f(set,)i(whic)m(h)e(is)f(due)i(to)e (the)h(b)s(oundedness)i(of)2586 3278 y Fl(~)2571 3302 y(A)e Fr(and)g(\(43\))o(.)62 b(The)83 3436 y(deriv)-5 b(ativ)m(es)33 b(of)g Fl(~)-57 b(q)739 3385 y Fx(\()p Fy(\006)p Fx(\))739 3463 y Fm(\030)885 3436 y Fr(are)33 b(therefore)g(b)s(ounded)g(on)g(the)g(compact)f(set)h Fo(K)7 b Fr(.)p 3250 3436 4 66 v 3254 3373 59 4 v 3254 3436 V 3312 3436 4 66 v 229 3624 a(W)-8 b(e)32 b(ma)m(y)f(no)m(w)h (state)g(the)g(\014rst)g(main)d(theorem,)j(whic)m(h)g(allo)m(ws)e(the)i (construction)83 3744 y(of)g Fo(L)260 3708 y Fx(2)332 3744 y Fr(w)m(a)m(v)m(e)j(pac)m(k)m(ets:)83 3926 y Fl(Theorem)i(2.)42 b Fp(L)-5 b(et)35 b Fo(\036)27 b Fn(\021)i Fr(\()p Fo(\036)1142 3941 y Fx(1)1181 3926 y Fo(;)17 b(:)g(:)g(:)e(;)i(\036)1457 3941 y Fx(4)1496 3926 y Fr(\))28 b Fn(2)g(C)1714 3889 y Fy(1)1708 3950 y Fx(0)1789 3926 y Fr(\()p Fj(R)1893 3889 y Fx(3)1938 3926 y Fr(\))1976 3889 y Fx(4)2016 3926 y Fp(.)188 4107 y(a\))49 b(The)34 b(wave)g(p)-5 b(acket)35 b Fo( )k Fp(de\014ne)-5 b(d)33 b(by)922 4392 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(=)1453 4325 y(1)p 1330 4369 294 4 v 1330 4461 a(\(2)p Fo(\031)t Fr(\))1514 4432 y Fx(3)p Fm(=)p Fx(2)1705 4268 y(4)1651 4298 y Fk(X)1661 4507 y Fm(r)r Fx(=1)1811 4257 y Fk(Z)1866 4482 y Fe(R)1914 4463 y Fg(3)1970 4392 y Fo(\036)2028 4407 y Fm(r)2065 4392 y Fr(\()p Fl(p)p Fr(\))p Fo( )2266 4407 y Fi(p)p Fm(;r)2370 4392 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)2639 4351 y Fx(3)2677 4392 y Fo(p)421 b Fr(\(47\))327 4669 y Fp(is)35 b(c)-5 b(ontinuous)34 b(in)h Fl(r)f Fp(and)h Fo(t)p Fp(.)193 4865 y(b\))49 b(F)-7 b(or)34 b(any)h Fo(n)28 b Fn(2)g Fj(N)942 4880 y Fx(0)987 4865 y Fp(,)35 b(ther)-5 b(e)35 b(is)f(a)h(c)-5 b(onstant)34 b Fo(c)1914 4880 y Fm(n)1996 4865 y Fp(such)h(that)473 5114 y Fn(jj)p Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fn(jj)853 5131 y Fe(C)893 5112 y Fg(4)965 5114 y Fn(\024)28 b Fo(c)1112 5129 y Fm(n)1250 5047 y Fr(1)21 b(+)h Fn(j)p Fo(t)p Fn(j)1509 5011 y Fm(n)p 1169 5091 467 4 v 1169 5183 a Fr(1)g(+)g Fn(j)p Fl(s)1410 5198 y Fm(t)1440 5183 y Fr(\()p Fl(r)p Fr(\))p Fn(j)1590 5154 y Fm(n)1674 5114 y Fr(=)27 b Fo(c)1819 5129 y Fm(n)2267 5047 y Fr(1)22 b(+)g Fn(j)p Fo(t)p Fn(j)2527 5011 y Fm(n)p 1876 5091 1089 4 v 1876 5202 a Fr(1)g(+)g([)p Fo(x)2127 5173 y Fx(2)2189 5202 y Fr(+)g Fo(y)2339 5173 y Fx(2)2400 5202 y Fr(+)g(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))2780 5173 y Fx(2)2820 5202 y Fr(])2847 5153 y Fm(n=)p Fx(2)2975 5114 y Fo(:)145 b Fr(\(48\))327 5388 y Fp(In)34 b(p)-5 b(articular)35 b Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)p Fr(\))27 b Fn(2)h Fo(L)1337 5352 y Fm(p)1377 5388 y Fr(\()p Fj(R)1481 5352 y Fx(3)1527 5388 y Fr(\))1565 5352 y Fx(4)1639 5388 y Fp(for)34 b(al)5 b(l)35 b Fo(t)28 b Fn(2)g Fj(R)46 b Fp(and)34 b Fo(p)28 b Fn(2)g Fr([1)p Fo(;)17 b Fn(1)p Fr(])p Fp(.)1678 5637 y Fr(9)p eop %%Page: 10 10 10 9 bop 432 573 a Fp(Pr)-5 b(o)g(of.)48 b(a\))26 b Fr(It)d(su\016ces)h (to)e(sho)m(w)i(the)f(result)f(for)g(the)h(summands)f Fo(r)k Fr(indep)s(enden)m(tly)-8 b(.)40 b(Let)432 693 y Fl(r)478 708 y Fm(n)552 693 y Fn(!)27 b Fl(r)f Fr(and)h Fo(t)970 708 y Fm(n)1045 693 y Fn(!)g Fo(t)p Fr(.)41 b(Then)28 b Fo( )1587 708 y Fi(p)p Fm(;r)1690 693 y Fr(\()p Fl(r)1774 708 y Fm(n)1821 693 y Fo(;)17 b(t)1900 708 y Fm(n)1947 693 y Fr(\))27 b Fn(!)h Fo( )2203 708 y Fi(p)p Fm(;r)2306 693 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))25 b(b)m(y)j(con)m(tin)m(uit)m(y)e(of)g(the)g(V)-8 b(olk)m(o)m(v)432 814 y(solutions)43 b(\(9\),)48 b(so)d(the)g(in)m(tegrand)f(in)g(\(47\)) g(con)m(v)m(erges)j(p)s(oin)m(t)m(wise.)80 b(F)-8 b(urthermore,)432 934 y(since)32 b Fo( )733 949 y Fi(p)p Fm(;r)836 934 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))31 b(is)h(b)s(ounded)g(in)f Fl(p)d Fn(2)g Fr(supp)35 b Fo(\036)2153 949 y Fm(r)2222 934 y Fr(as)d(w)m(ell)f(as)h(in)f Fl(r)h Fr(and)g Fo(t)p Fr(,)g(con)m(tin)m(uit)m(y)g(of)432 1054 y Fo( )k Fr(is)c(established)h (b)m(y)g(dominated)f(con)m(v)m(ergence,)j(see,)f(e.)f(g.,)f(Theorem)h (I.11)g(in.)3450 1018 y Fx(25)578 1186 y Fp(b\))j Fr(W)-8 b(e)33 b(will)e(sho)m(w)j(that)f(for)f(an)m(y)h Fo(n)1897 1201 y Fm(x)1941 1186 y Fo(;)17 b(n)2043 1201 y Fm(y)2085 1186 y Fo(;)g(n)2187 1201 y Fm(z)2255 1186 y Fn(2)28 b Fj(N)2415 1201 y Fx(0)2461 1186 y Fr(,)k(there)i(is)e(a)h(constan)m (t)h Fo(c)3386 1201 y Fm(n)3429 1209 y Fh(x)3467 1201 y Fm(;n)3530 1209 y Fh(y)3567 1201 y Fm(;n)3630 1209 y Fh(z)432 1306 y Fr(suc)m(h)g(that)840 1560 y Fn(j)p Fo(x)p Fn(j)951 1519 y Fm(n)994 1527 y Fh(x)1036 1560 y Fn(j)p Fo(y)t Fn(j)1144 1519 y Fm(n)1187 1527 y Fh(y)1227 1560 y Fn(j)p Fo(z)27 b Fn(\000)22 b Fo(t)p Fn(j)1489 1519 y Fm(n)1532 1527 y Fh(z)1572 1560 y Fn(jj)p Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fn(jj)1952 1577 y Fe(C)1992 1558 y Fg(4)2065 1560 y Fn(\024)28 b Fo(c)2212 1575 y Fm(n)2255 1583 y Fh(x)2294 1575 y Fm(;n)2357 1583 y Fh(y)2394 1575 y Fm(;n)2457 1583 y Fh(z)2513 1479 y Fk(\000)2558 1560 y Fr(1)22 b(+)g Fn(j)p Fo(t)p Fn(j)2818 1519 y Fm(n)2861 1527 y Fh(x)2899 1519 y Fx(+)p Fm(n)2997 1527 y Fh(y)3035 1519 y Fx(+)p Fm(n)3133 1527 y Fh(z)3172 1479 y Fk(\001)3235 1560 y Fo(:)234 b Fr(\(49\))432 1814 y(The)33 b(estimate)f(in)g(\(48\))g(follo)m(ws)f(from)g(this)i(b)m(y)g (using)531 2041 y Fk(\002)573 2122 y Fo(x)628 2081 y Fx(2)690 2122 y Fr(+)22 b Fo(y)840 2081 y Fx(2)901 2122 y Fr(+)g(\()p Fo(z)k Fn(\000)d Fo(t)p Fr(\))1281 2081 y Fx(2)1321 2041 y Fk(\003)1362 2064 y Fm(n=)p Fx(2)1507 2122 y Fr(=)1611 2011 y Fk(\020)1670 2032 y(p)p 1770 2032 748 4 v 90 x Fo(x)1825 2093 y Fx(2)1887 2122 y Fr(+)f Fo(y)2037 2093 y Fx(2)2098 2122 y Fr(+)g(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))2478 2093 y Fx(2)2518 2011 y Fk(\021)2577 2034 y Fm(n)2652 2122 y Fn(\024)1267 2226 y Fk(\022)1341 2239 y(q)p 1440 2239 835 4 v 1440 2367 a Fr(\()p Fn(j)p Fo(x)p Fn(j)g Fr(+)g Fn(j)p Fo(y)t Fn(j)f Fr(+)h Fn(j)p Fo(z)k Fn(\000)d Fo(t)p Fn(j)p Fr(\))2236 2318 y Fx(2)2275 2226 y Fk(\023)2349 2249 y Fm(n)2423 2367 y Fr(=)28 b(\()p Fn(j)p Fo(x)p Fn(j)22 b Fr(+)g Fn(j)p Fo(y)t Fn(j)e Fr(+)i Fn(j)p Fo(z)27 b Fn(\000)c Fo(t)p Fn(j)p Fr(\))3322 2318 y Fm(n)3386 2367 y Fo(;)83 b Fr(\(50\))432 2678 y(where)33 b(the)g(righ)m(t-hand)e(side)i(is)e(a)h (p)s(olynomial)d(of)j(degree)h Fo(n)g Fr(in)e Fn(j)p Fo(x)p Fn(j)p Fr(,)h Fn(j)p Fo(y)t Fn(j)p Fr(,)f(and)i Fn(j)p Fo(z)26 b Fn(\000)c Fo(t)p Fn(j)p Fr(.)578 2809 y(In)36 b(order)f(to)g(pro)m(v)m(e)h(the)g(estimate)f(\(49\),)g(w)m(e)h (shall)e(apply)h(the)h(stationary-phase)432 2930 y(metho)s(d)29 b(\(see,)i(for)e(example,)h(Theorem)g(XI.14)g(in)2311 2894 y Fx(26)2385 2930 y Fr(\).)42 b(W)-8 b(e)30 b(only)f(need)i(to)e (lo)s(ok)g(at)g(the)432 3050 y(case)37 b Fl(r)d Fn(\021)g Fr(\()p Fo(x;)17 b(y)t(;)g(z)t Fr(\))34 b Fn(2)h Fj(R)1354 3014 y Fx(3)1435 3050 y Fr(with)h Fo(x;)17 b(y)t(;)g(z)29 b Fn(\000)c Fo(t)34 b Fn(6)p Fr(=)g(0;)39 b(for)c Fo(x)p Fr(,)j Fo(y)t Fr(,)f(or)f Fo(z)29 b Fn(\000)c Fo(t)37 b Fr(equal)f(to)g(zero,)432 3170 y(\(49\))c(follo)m(ws)f(b)m(y)i(con)m (tin)m(uit)m(y)-8 b(.)44 b(Setting)32 b Fo(\017)c Fr(:=)g Fo(\017)2157 3185 y Fm(r)2227 3170 y Fr(for)k(abbreviation,)g(w)m(e)i (\014nd)538 3337 y Fk(Z)594 3563 y Fe(R)642 3544 y Fg(3)697 3473 y Fo(\036)755 3488 y Fm(r)793 3473 y Fr(\()p Fl(p)p Fr(\))p Fo( )994 3488 y Fi(p)p Fm(;r)1097 3473 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)1366 3431 y Fx(3)1404 3473 y Fo(p)621 3731 y Fr(=)780 3596 y Fk(Z)835 3821 y Fe(R)883 3802 y Fg(3)939 3731 y Fo(\036)997 3746 y Fm(r)1034 3731 y Fr(\()p Fl(p)p Fr(\))p Fl(M)1278 3690 y Fx(\()p Fm(\017)p Fx(\))1366 3731 y Fr(\()p Fl(p)p Fo(;)g(t)23 b Fn(\000)f Fo(z)t Fr(\))p Fo(u)1810 3746 y Fm(r)1849 3731 y Fr(\()p Fl(p)p Fr(\))17 b(exp)2169 3621 y Fk(h)2216 3731 y Fo(i)2266 3621 y Fk(\020)2325 3731 y Fl(q)2384 3680 y Fx(\()p Fm(\017)p Fx(\))2384 3754 y Fm(t)p Fy(\000)p Fm(z)2505 3731 y Fr(\()p Fl(p)p Fr(\))22 b Fn(\001)g Fl(s)2759 3746 y Fm(t)2789 3731 y Fr(\()p Fl(r)p Fr(\))f Fn(\000)i Fo(\017\024)p Fr(\()p Fl(p)p Fr(\))p Fo(t)3300 3621 y Fk(\021)q(i)3424 3731 y Fo(d)3475 3690 y Fx(3)3514 3731 y Fo(p)621 3990 y Fr(=)780 3854 y Fk(Z)835 4080 y Fe(R)883 4061 y Fg(3)939 3990 y Fo(\036)997 4005 y Fm(r)1034 3990 y Fr(\()r Fl(~)-58 b(q)1131 3939 y Fx(\()p Fm(\017)p Fx(\))1131 4013 y Fm(t)p Fy(\000)p Fm(z)1252 3990 y Fr(\()s Fl(~)f(p)p Fr(\)\))p Fl(M)1534 3949 y Fx(\()p Fm(\017)p Fx(\))1622 3990 y Fr(\()r Fl(~)h(q)1719 3939 y Fx(\()p Fm(\017)p Fx(\))1719 4013 y Fm(t)p Fy(\000)p Fm(z)1839 3990 y Fr(\()s Fl(~)f(p)p Fr(\))p Fo(;)17 b(t)22 b Fn(\000)h Fo(z)t Fr(\))p Fo(u)2321 4005 y Fm(r)2359 3990 y Fr(\()r Fl(~)-58 b(q)2456 3939 y Fx(\()p Fm(\017)p Fx(\))2456 4013 y Fm(t)p Fy(\000)p Fm(z)2576 3990 y Fr(\()s Fl(~)f(p)p Fr(\)\))780 4228 y Fn(\002)17 b Fr(exp)1040 4117 y Fk(h)1087 4228 y Fo(i)1137 4117 y Fk(\020)1199 4228 y Fl(~)-59 b(p)23 b Fn(\001)e Fl(s)1374 4243 y Fm(t)1404 4228 y Fr(\()p Fl(r)p Fr(\))h Fn(\000)h Fo(\017\024)p Fr(\()r Fl(~)-58 b(q)1840 4177 y Fx(\()p Fm(\017)p Fx(\))1840 4250 y Fm(t)p Fy(\000)p Fm(z)1961 4228 y Fr(\()s Fl(~)f(p)p Fr(\)\))p Fo(t)2172 4117 y Fk(\021i)2295 4113 y(\014)2295 4173 y(\014)2295 4233 y(\014)2328 4228 y Fr(det)2480 4117 y Fk(\020)2540 4228 y Fo(D)t Fl(~)i(q)2683 4177 y Fx(\()p Fm(\017)p Fx(\))2683 4250 y Fm(t)p Fy(\000)p Fm(z)2803 4228 y Fr(\()s Fl(~)e(p)p Fr(\))2941 4117 y Fk(\021)3000 4113 y(\014)3000 4173 y(\014)3000 4233 y(\014)3050 4228 y Fo(d)3101 4187 y Fx(3)3149 4228 y Fr(~)h Fo(p)621 4469 y Fr(=)841 4402 y(\()p Fn(\000)p Fo(i)p Fr(\))1027 4366 y Fm(n)1070 4374 y Fh(x)1109 4366 y Fx(+)p Fm(n)1207 4374 y Fh(y)1244 4366 y Fx(+)p Fm(n)1342 4374 y Fh(z)p 790 4447 643 4 v 790 4538 a Fo(x)845 4509 y Fm(n)888 4517 y Fh(x)931 4538 y Fo(y)983 4509 y Fm(n)1026 4517 y Fh(y)1067 4538 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))1349 4509 y Fm(n)1392 4517 y Fh(z)1459 4334 y Fk(Z)1514 4559 y Fe(R)1562 4541 y Fg(3)1618 4469 y Fo(\036)1676 4484 y Fm(r)1713 4469 y Fr(\()r Fl(~)-58 b(q)1810 4419 y Fx(\()p Fm(\017)p Fx(\))1810 4492 y Fm(t)p Fy(\000)p Fm(z)1930 4469 y Fr(\()s Fl(~)f(p)q Fr(\)\))p Fl(M)2213 4428 y Fx(\()p Fm(\017)p Fx(\))2300 4469 y Fr(\()r Fl(~)h(q)2397 4419 y Fx(\()p Fm(\017)p Fx(\))2397 4492 y Fm(t)p Fy(\000)p Fm(z)2518 4469 y Fr(\()s Fl(~)f(p)p Fr(\))p Fo(;)17 b(t)22 b Fn(\000)h Fo(z)t Fr(\))p Fo(u)3000 4484 y Fm(r)3038 4469 y Fr(\()r Fl(~)-58 b(q)3135 4419 y Fx(\()p Fm(\017)p Fx(\))3135 4492 y Fm(t)p Fy(\000)p Fm(z)3255 4469 y Fr(\()s Fl(~)f(p)p Fr(\)\))780 4710 y Fn(\002)17 b Fr(exp)1040 4600 y Fk(\020)1099 4710 y Fn(\000)p Fo(i\017\024)p Fr(\()r Fl(~)-58 b(q)1401 4660 y Fx(\()p Fm(\017)p Fx(\))1401 4733 y Fm(t)p Fy(\000)p Fm(z)1523 4710 y Fr(\()s Fl(~)f(p)p Fr(\)\))p Fo(t)1734 4600 y Fk(\021)1810 4596 y(\014)1810 4656 y(\014)1810 4715 y(\014)1843 4710 y Fr(det)1996 4600 y Fk(\020)2055 4710 y Fo(D)t Fl(~)i(q)2198 4660 y Fx(\()p Fm(\017)p Fx(\))2198 4733 y Fm(t)p Fy(\000)p Fm(z)2318 4710 y Fr(\()s Fl(~)e(p)p Fr(\))2456 4600 y Fk(\021)2516 4596 y(\014)2516 4656 y(\014)2516 4715 y(\014)780 4955 y Fn(\002)874 4815 y Fk(\024\022)1056 4888 y Fo(@)p 1010 4932 150 4 v 1010 5024 a(@)14 b Fr(~)-58 b Fo(p)1115 5039 y Fm(x)1170 4815 y Fk(\023)1243 4837 y Fm(n)1286 4845 y Fh(x)1346 4815 y Fk(\022)1474 4888 y Fo(@)p 1429 4932 148 4 v 1429 5024 a(@)14 b Fr(~)-58 b Fo(p)1534 5039 y Fm(y)1586 4815 y Fk(\023)1659 4837 y Fm(n)1702 4845 y Fh(y)1761 4815 y Fk(\022)1888 4888 y Fo(@)p 1844 4932 146 4 v 1844 5024 a(@)14 b Fr(~)-58 b Fo(p)1949 5039 y Fm(z)1999 4815 y Fk(\023)2073 4837 y Fm(n)2116 4845 y Fh(z)2172 4955 y Fr(exp)18 b(\()p Fo(i)s Fl(~)-59 b(p)22 b Fn(\001)g Fl(s)2587 4970 y Fm(t)2617 4955 y Fr(\()p Fl(r)p Fr(\)\))2777 4815 y Fk(\025)2846 4955 y Fo(d)2897 4914 y Fx(3)2944 4955 y Fr(~)-57 b Fo(p;)484 b Fr(\(51\))432 5268 y(where)39 b(\(42\))e(has)i(b)s(een)f(used)h(and)g (the)f(transformation)e(form)m(ula)g(for)h(in)m(tegrals)g(has)432 5388 y(b)s(een)g(applied.)56 b(Since)37 b Fo(\036)1376 5403 y Fm(r)1449 5388 y Fn(2)e(C)1608 5352 y Fy(1)1602 5413 y Fx(0)1720 5388 y Fr(and)i(all)e(other)40 b Fl(~)-59 b(p)37 b Fr(dep)s(endences)j(are)d Fn(C)3203 5352 y Fy(1)3278 5388 y Fr(,)h(w)m(e)g(ma)m(y)2002 5637 y(10)p eop %%Page: 11 11 11 10 bop 83 573 a Fr(p)s(erform)32 b(in)m(tegrations)f(b)m(y)i(parts)g (and)g(obtain)229 711 y Fk(Z)284 936 y Fe(R)332 917 y Fg(3)388 846 y Fo(\036)446 861 y Fm(r)483 846 y Fr(\()p Fl(p)p Fr(\))p Fo( )684 861 y Fi(p)p Fm(;r)788 846 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)1057 805 y Fx(3)1095 846 y Fo(p)312 1116 y Fr(=)608 1049 y Fo(i)641 1013 y Fm(n)684 1021 y Fh(x)723 1013 y Fx(+)p Fm(n)821 1021 y Fh(y)858 1013 y Fx(+)p Fm(n)956 1021 y Fh(z)p 481 1093 643 4 v 481 1185 a Fo(x)536 1156 y Fm(n)579 1164 y Fh(x)622 1185 y Fo(y)674 1156 y Fm(n)717 1164 y Fh(y)758 1185 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))1040 1156 y Fm(n)1083 1164 y Fh(z)1150 981 y Fk(Z)1205 1206 y Fe(R)1253 1187 y Fg(3)1308 1116 y Fr(exp)c(\()p Fo(i)s Fl(~)-59 b(p)23 b Fn(\001)e Fl(s)1723 1131 y Fm(t)1753 1116 y Fr(\()p Fl(r)p Fr(\)\))1930 976 y Fk(\022)2059 1049 y Fo(@)p 2013 1093 150 4 v 2013 1185 a(@)14 b Fr(~)-58 b Fo(p)2118 1200 y Fm(x)2173 976 y Fk(\023)2246 998 y Fm(n)2289 1006 y Fh(x)2348 976 y Fk(\022)2477 1049 y Fo(@)p 2432 1093 148 4 v 2432 1185 a(@)14 b Fr(~)-58 b Fo(p)2537 1200 y Fm(y)2589 976 y Fk(\023)2662 998 y Fm(n)2705 1006 y Fh(y)2763 976 y Fk(\022)2891 1049 y Fo(@)p 2847 1093 146 4 v 2847 1185 a(@)14 b Fr(~)-58 b Fo(p)2952 1200 y Fm(z)3002 976 y Fk(\023)3076 998 y Fm(n)3119 1006 y Fh(z)471 1250 y Fk(\024)523 1390 y Fo(\036)581 1405 y Fm(r)619 1390 y Fr(\()r Fl(~)g(q)716 1339 y Fx(\()p Fm(\017)p Fx(\))716 1413 y Fm(t)p Fy(\000)p Fm(z)836 1390 y Fr(\()s Fl(~)f(p)q Fr(\)\))p Fl(M)1119 1349 y Fx(\()p Fm(\017)p Fx(\))1206 1390 y Fr(\()r Fl(~)h(q)1303 1339 y Fx(\()p Fm(\017)p Fx(\))1303 1413 y Fm(t)p Fy(\000)p Fm(z)1424 1390 y Fr(\()s Fl(~)f(p)p Fr(\))p Fo(;)17 b(t)22 b Fn(\000)g Fo(z)t Fr(\))p Fo(u)1905 1405 y Fm(r)1944 1390 y Fr(\()r Fl(~)-58 b(q)2041 1339 y Fx(\()p Fm(\017)p Fx(\))2041 1413 y Fm(t)p Fy(\000)p Fm(z)2161 1390 y Fr(\()s Fl(~)f(p)p Fr(\)\))471 1662 y Fn(\002)17 b Fr(exp)731 1552 y Fk(\020)790 1662 y Fn(\000)p Fo(i\017\024)p Fr(\()r Fl(~)-58 b(q)1092 1611 y Fx(\()p Fm(\017)p Fx(\))1092 1685 y Fm(t)p Fy(\000)p Fm(z)1214 1662 y Fr(\()s Fl(~)f(p)p Fr(\)\))p Fo(t)1425 1552 y Fk(\021)1501 1548 y(\014)1501 1608 y(\014)1501 1667 y(\014)1534 1662 y Fr(det)1686 1552 y Fk(\020)1746 1662 y Fo(D)t Fl(~)i(q)1889 1611 y Fx(\()p Fm(\017)p Fx(\))1889 1685 y Fm(t)p Fy(\000)p Fm(z)2009 1662 y Fr(\()s Fl(~)e(p)p Fr(\))2147 1552 y Fk(\021)2207 1548 y(\014)2207 1608 y(\014)2207 1667 y(\014)2256 1522 y(\025)2326 1662 y Fo(d)2377 1621 y Fx(3)2424 1662 y Fr(~)i Fo(p:)655 b Fr(\(52\))83 1945 y(Observ)m(e)35 b(that)d(b)m(y)h(the)g(c)m(hain)g(rule,)183 2110 y Fk(\014)183 2169 y(\014)183 2229 y(\014)183 2289 y(\014)216 2114 y(\022)346 2187 y Fo(@)p 299 2231 150 4 v 299 2322 a(@)14 b Fr(~)-58 b Fo(p)404 2337 y Fm(x)459 2114 y Fk(\023)532 2136 y Fm(m)594 2144 y Fh(x)654 2114 y Fk(\022)783 2187 y Fo(@)p 738 2231 148 4 v 738 2322 a(@)14 b Fr(~)-58 b Fo(p)843 2337 y Fm(y)895 2114 y Fk(\023)968 2136 y Fm(m)1030 2144 y Fh(y)1089 2114 y Fk(\022)1217 2187 y Fo(@)p 1172 2231 146 4 v 1172 2322 a(@)14 b Fr(~)-58 b Fo(p)1277 2337 y Fm(z)1328 2114 y Fk(\023)1401 2136 y Fm(m)1463 2144 y Fh(z)1520 2254 y Fr(exp)1686 2144 y Fk(\020)1745 2254 y Fn(\000)p Fo(i\017\024)p Fr(\()r Fl(~)g(q)2047 2203 y Fx(\()p Fm(\017)p Fx(\))2047 2277 y Fm(t)p Fy(\000)p Fm(z)2169 2254 y Fr(\()s Fl(~)f(p)p Fr(\)\))p Fo(t)2380 2144 y Fk(\021)2440 2110 y(\014)2440 2169 y(\014)2440 2229 y(\014)2440 2289 y(\014)2500 2254 y Fn(\024)1206 2468 y Fo(d)1257 2483 y Fm(m)1319 2491 y Fh(x)1358 2483 y Fm(;m)1440 2491 y Fh(y)1477 2483 y Fm(;m)1559 2491 y Fh(z)1616 2387 y Fk(\000)1661 2468 y Fr(1)22 b(+)g Fn(j)p Fo(t)p Fn(j)1921 2427 y Fm(m)1983 2435 y Fh(x)2022 2427 y Fx(+)p Fm(m)2139 2435 y Fh(y)2177 2427 y Fx(+)p Fm(m)2294 2435 y Fh(z)2334 2387 y Fk(\001)2424 2468 y Fo(;)48 b Fl(~)-59 b(p)28 b Fn(2)g Fr(supp)34 b Fo(\036)2972 2483 y Fm(r)3037 2468 y Fo(;)83 b Fr(\(53\))83 2699 y(for)31 b(some)h(constan)m(ts)h Fo(d)957 2714 y Fm(m)1019 2722 y Fh(x)1058 2714 y Fm(;m)1140 2722 y Fh(y)1178 2714 y Fm(;m)1260 2722 y Fh(z)1300 2699 y Fr(.)43 b(Again)31 b(b)m(y)h(the)h(c)m(hain)e(rule)h(and)f(b)m(y)i(Lemma)e(1,)g(the)83 2819 y(deriv)-5 b(ativ)m(es)39 b(in)g(\(52\))g(are)g(uniformly)e(b)s (ounded)k(in)d Fl(r)p Fr(,)j(and)f(w)m(e)g(ma)m(y)f(conclude)h(that)83 2939 y(the)35 b(in)m(tegral)d(is)i(b)s(ounded)h(in)f Fl(r)g Fr(and)g(that)g(the)h Fo(t)f Fr(dep)s(endence)j(is)c(go)m(v)m (erned)j(b)m(y)f(\(53\).)83 3060 y(Th)m(us,)145 3342 y Fn(j)p Fo(x)p Fn(j)256 3301 y Fm(n)299 3309 y Fh(x)342 3342 y Fn(j)p Fo(y)t Fn(j)450 3301 y Fm(n)493 3309 y Fh(y)533 3342 y Fn(j)p Fo(z)26 b Fn(\000)d Fo(t)p Fn(j)795 3301 y Fm(n)838 3309 y Fh(z)894 3198 y Fk(\014)894 3257 y(\014)894 3317 y(\014)894 3377 y(\014)927 3198 y(\014)927 3257 y(\014)927 3317 y(\014)927 3377 y(\014)961 3207 y(Z)1016 3432 y Fe(R)1064 3413 y Fg(3)1119 3342 y Fo(\036)1177 3357 y Fm(r)1215 3342 y Fr(\()p Fl(p)p Fr(\))p Fo( )1416 3357 y Fi(p)p Fm(;r)1519 3342 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)1788 3301 y Fx(3)1826 3342 y Fo(p)1875 3198 y Fk(\014)1875 3257 y(\014)1875 3317 y(\014)1875 3377 y(\014)1909 3198 y(\014)1909 3257 y(\014)1909 3317 y(\014)1909 3377 y(\014)1942 3441 y Fe(C)1984 3422 y Fg(4)2056 3342 y Fn(\024)30 b Fr(~)-51 b Fo(c)2203 3357 y Fm(r)n(;n)2296 3365 y Fh(x)2335 3357 y Fm(;n)2398 3365 y Fh(y)2435 3357 y Fm(;n)2498 3365 y Fh(z)2553 3261 y Fk(\000)2599 3342 y Fr(1)22 b(+)g Fn(j)p Fo(t)p Fn(j)2859 3301 y Fm(n)2902 3309 y Fh(x)2940 3301 y Fx(+)p Fm(n)3038 3309 y Fh(y)3076 3301 y Fx(+)p Fm(n)3174 3309 y Fh(z)3213 3261 y Fk(\001)3147 3524 y Fr(\(54\))83 3644 y(for)j(some)g(constan)m (ts)k(~)-51 b Fo(c)929 3659 y Fm(r)n(;n)1022 3667 y Fh(x)1061 3659 y Fm(;n)1124 3667 y Fh(y)1161 3659 y Fm(;n)1224 3667 y Fh(z)1263 3644 y Fr(,)27 b(from)d(whic)m(h)i(\(49\))f(follo)m (ws)g(after)g(summation)e(o)m(v)m(er)k Fo(r)s Fr(.)p 3250 3765 4 66 v 3254 3702 59 4 v 3254 3765 V 3312 3765 4 66 v 229 3991 a(Theorem)33 b(2)g(allo)m(ws)e(us)i(to)f(de\014ne)i (regular)e(w)m(a)m(v)m(e)i(pac)m(k)m(ets:)83 4212 y Fl(De\014nition)i (1.)42 b Fp(F)-7 b(or)34 b Fo(t)28 b Fn(2)g Fj(R)5 b Fp(,)40 b(de\014ne)667 4443 y Fn(F)739 4458 y Fm(t)796 4443 y Fr(:)28 b Fn(C)909 4402 y Fy(1)903 4467 y Fx(0)984 4443 y Fr(\()p Fj(R)1088 4402 y Fx(3)1133 4443 y Fr(\))1171 4402 y Fx(4)1238 4443 y Fn(!)g Fo(L)1432 4402 y Fx(2)1472 4443 y Fr(\()p Fj(R)1575 4402 y Fx(3)1621 4443 y Fr(\))1659 4402 y Fx(4)1720 4443 y Fn(\\)23 b(C)1861 4458 y Fy(1)1936 4443 y Fr(\()p Fj(R)2040 4402 y Fx(3)2085 4443 y Fr(\))2123 4402 y Fx(4)2190 4443 y Fo(;)45 b(\036)27 b Fn(7!)h(F)2547 4458 y Fm(t)2576 4443 y Fr(\()p Fo(\036)p Fr(\))p Fo(;)410 b Fr(\(55\))83 4673 y Fp(wher)-5 b(e)696 4869 y Fr(\()o Fn(F)805 4884 y Fm(t)835 4869 y Fr(\()p Fo(\036)p Fr(\)\))16 b(\()p Fl(r)p Fr(\))27 b(:=)1435 4802 y(1)p 1313 4846 294 4 v 1313 4938 a(\(2)p Fo(\031)t Fr(\))1497 4909 y Fx(3)p Fm(=)p Fx(2)1687 4745 y(4)1633 4775 y Fk(X)1643 4984 y Fm(r)r Fx(=1)1793 4734 y Fk(Z)1849 4959 y Fe(R)1897 4940 y Fg(3)1952 4869 y Fo(\036)2010 4884 y Fm(r)2048 4869 y Fr(\()p Fl(p)p Fr(\))p Fo( )2249 4884 y Fi(p)p Fm(;r)2352 4869 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)2621 4828 y Fx(3)2659 4869 y Fo(p)439 b Fr(\(56\))83 5147 y Fp(and)40 b Fn(C)330 5162 y Fy(1)405 5147 y Fr(\()p Fj(R)509 5111 y Fx(3)554 5147 y Fr(\))592 5111 y Fx(4)669 5147 y Fn(\032)e(C)836 5162 y Fm(b)871 5147 y Fr(\()p Fj(R)975 5111 y Fx(3)1020 5147 y Fr(\))1058 5111 y Fx(4)1138 5147 y Fp(is)i(the)g(sp)-5 b(ac)g(e)39 b(of)h(c)-5 b(ontinuous)40 b(functions)g Fj(R)2785 5111 y Fx(3)2868 5147 y Fn(!)d Fj(C)3071 5111 y Fx(4)3156 5147 y Fp(that)83 5268 y(vanish)47 b(at)h(in\014nity.)84 b(The)47 b(elements)g(of)h(the)g(r)-5 b(ange)47 b Fn(F)2227 5283 y Fm(t)2273 5268 y Fr(\()p Fn(C)2369 5232 y Fy(1)2363 5292 y Fx(0)2444 5268 y Fr(\()p Fj(R)2548 5232 y Fx(3)2593 5268 y Fr(\))2631 5232 y Fx(4)2671 5268 y Fr(\))h Fp(wil)5 b(l)47 b(b)-5 b(e)48 b(c)-5 b(al)5 b(le)-5 b(d)83 5388 y Fr(regular)32 b(\(V)-8 b(olk)m(o)m(v\))32 b(w)m(a)m(v)m(e)j(pac)m(k)m(ets)p Fp(.)1653 5637 y Fr(11)p eop %%Page: 12 12 12 11 bop 578 573 a Fr(The)42 b(follo)m(wing)d(theorem)i(assures)i (that)e(under)h(w)m(eak)h(assumptions,)g(the)f(w)m(a)m(v)m(e)432 693 y(pac)m(k)m(ets)36 b(de\014ned)g(ab)s(o)m(v)m(e)f(actually)e (ful\014ll)e(the)k(Dirac)e(equation)h(in)g(the)g(Hamiltonian)432 814 y(form)e(\(30\).)47 b(W)-8 b(e)34 b(only)f(need)i(the)f(additional) d(requiremen)m(t)j(that)2925 790 y Fl(~)2911 814 y(A)29 b Fn(2)h(C)3179 777 y Fx(1)3173 840 y Fm(b)3219 814 y Fr(,)k(i.)f(e.)3500 790 y Fl(~)3486 814 y(A)g Fr(is)432 934 y(con)m(tin)m(uously)g(di\013eren)m(tiable)e(and)i(b)s(oth)2018 910 y Fl(~)2004 934 y(A)f Fr(and)h(the)g(deriv)-5 b(ativ)m(e)2940 910 y Fl(~)2926 934 y(A)3011 898 y Fy(0)3066 934 y Fr(are)33 b(b)s(ounded.)432 1130 y Fl(Theorem)k(3.)k Fp(L)-5 b(et)36 b Fo(\036)27 b Fn(2)h(C)1441 1093 y Fy(1)1435 1154 y Fx(0)1516 1130 y Fr(\()p Fj(R)1620 1093 y Fx(3)1665 1130 y Fr(\))1703 1093 y Fx(4)1743 1130 y Fp(,)35 b(and)f(set)1646 1340 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(:=)h Fn(F)2144 1355 y Fm(t)2173 1340 y Fr(\()p Fo(\036)p Fr(\)\()p Fl(r)p Fr(\))p Fo(:)1040 b Fr(\(57\))432 1551 y Fp(Assume)35 b(that)1009 1527 y Fl(~)995 1551 y(A)27 b Fn(2)h(C)1259 1515 y Fx(1)1253 1577 y Fm(b)1299 1551 y Fp(.)45 b(Then)34 b(the)h(fol)5 b(lowing)33 b(assertions)h(hold:)537 1747 y(a\))49 b Fo( )31 b Fn(2)d(C)922 1710 y Fx(1)962 1747 y Fr(\()p Fj(R)1066 1710 y Fx(4)1111 1747 y Fr(\))1149 1710 y Fx(4)1224 1747 y Fp(and)804 1985 y Fo(i@)888 2000 y Fm(t)919 1985 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(=)h Fo(H)8 b Fr(\()p Fo(t)p Fr(\))p Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))26 b Fn(\021)1918 1874 y Fk(h)1965 1985 y Ff(\013)2057 1874 y Fk(\020)2116 1985 y Fn(\000)p Fo(i)p Fd(r)e Fn(\000)e Fo(e)2503 1961 y Fl(~)2489 1985 y(A)p Fr(\()p Fo(t)h Fn(\000)f Fo(z)t Fr(\))2856 1874 y Fk(\021)2939 1985 y Fr(+)g Fo(\014)6 b(m)3183 1874 y Fk(i)3246 1985 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fo(:)3496 2133 y Fr(\(58\))542 2333 y Fp(b\))49 b(Ther)-5 b(e)34 b(ar)-5 b(e)34 b(c)-5 b(onstants)35 b Fo(c)1592 2348 y Fm(n)1639 2333 y Fp(,)g Fo(n)27 b Fn(2)i Fj(N)1949 2348 y Fx(0)1995 2333 y Fp(,)35 b(satisfying)1107 2599 y Fn(jj)o Fr(\()p Fo(H)8 b Fr(\()p Fo(t)p Fr(\))p Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)p Fr(\)\))f(\()p Fl(r)p Fr(\))p Fn(j)o(j)28 b(\024)g Fo(c)2056 2614 y Fm(n)2504 2532 y Fr(1)22 b(+)g Fn(j)p Fo(t)p Fn(j)2764 2496 y Fm(n)p 2113 2576 1089 4 v 2113 2686 a Fr(1)g(+)g([)p Fo(x)2364 2658 y Fx(2)2426 2686 y Fr(+)g Fo(y)2576 2658 y Fx(2)2636 2686 y Fr(+)g(\()p Fo(z)27 b Fn(\000)c Fo(t)p Fr(\))3017 2658 y Fx(2)3056 2686 y Fr(])3084 2638 y Fm(n=)p Fx(2)3211 2599 y Fo(:)258 b Fr(\(59\))676 2889 y Fp(Ther)-5 b(efor)g(e,)33 b Fo(H)8 b Fr(\()p Fo(t)p Fr(\))p Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)p Fr(\))27 b Fn(2)h Fo(L)1783 2853 y Fx(2)1823 2889 y Fr(\()p Fj(R)1927 2853 y Fx(3)1972 2889 y Fr(\))2010 2853 y Fx(4)2085 2889 y Fp(for)34 b(al)5 b(l)35 b Fo(t)p Fp(.)542 3090 y(c\))49 b(The)34 b(mapping)f Fj(R)39 b Fn(!)27 b Fo(L)1561 3054 y Fx(2)1601 3090 y Fr(\()p Fj(R)1705 3054 y Fx(3)1750 3090 y Fr(\))1788 3054 y Fx(4)1828 3090 y Fo(;)33 b(t)28 b Fn(7!)f Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)p Fr(\))p Fo(;)35 b Fp(is)f Fn(C)2552 3054 y Fx(1)2592 3090 y Fp(,)h(and)1675 3350 y Fo(i)1735 3283 y(d)p 1718 3327 86 4 v 1718 3418 a(dt)1814 3350 y( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)p Fr(\))27 b(=)g Fo(H)8 b Fr(\()p Fo(t)p Fr(\))p Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)p Fr(\))p Fo(;)825 b Fr(\(60\))676 3596 y Fp(wher)-5 b(e)34 b(the)h(derivative)f(is)g(taken)h(in)f(the)h(sp)-5 b(ac)g(e)34 b Fo(L)2527 3560 y Fx(2)2567 3596 y Fr(\()p Fj(R)2671 3560 y Fx(3)2717 3596 y Fr(\))2755 3560 y Fx(4)2794 3596 y Fp(.)432 3792 y(Pr)-5 b(o)g(of.)48 b(a\))37 b Fr(Since)c(the)h(V)-8 b(olk)m(o)m(v)33 b(solutions)f Fo( )2079 3807 y Fi(p)p Fm(;r)2216 3792 y Fr(are)h Fn(C)2437 3755 y Fx(1)2510 3792 y Fr(and)g(satisfy)h(the)f(Dirac)f(equa-)432 3912 y(tion)45 b(\(30\))o(,)50 b(w)m(e)e(only)e(ha)m(v)m(e)h(to)f(sho)m (w)i(that)e(the)h(di\013eren)m(tiations)e(in)g(\(58\))h(ma)m(y)g(b)s(e) 432 4032 y(in)m(terc)m(hanged)36 b(with)f(the)g(in)m(tegration)f(in)h (\(56\))o(.)51 b(This)36 b(is)f(p)s(ossible)f(since)i(the)g(deriv)-5 b(a-)432 4153 y(tiv)m(es)30 b(of)f Fo( )828 4168 y Fi(p)p Fm(;r)932 4153 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))29 b(with)g(resp)s(ect)i(to)f Fo(t)p Fr(,)g Fo(x)p Fr(,)h Fo(y)t Fr(,)f(and)f Fo(z)35 b Fr(are)30 b(con)m(tin)m(uous)g(functions) g(of)f Fl(p)p Fr(,)432 4273 y Fl(r)p Fr(,)j(and)h Fo(t)g Fr(and)f(since)h(the)g(supp)s(ort)g(of)f Fo(\036)g Fr(is)h(compact.)578 4393 y Fp(b\))56 b Fr(Up)s(on)c(in)m(terc)m(hange)h(of)f(di\013eren)m (tiation)f(and)i(in)m(tegration)d(and)j(using)f(the)432 4514 y(b)s(oundedness)40 b(of)f(the)f(deriv)-5 b(ativ)m(e)1769 4490 y Fl(~)1755 4514 y(A)1840 4478 y Fy(0)1862 4514 y Fr(,)41 b(this)d(statemen)m(t)h(follo)m(ws)e(as)i(in)e(the)i(pro)s (of)f(of)432 4634 y(Theorem)33 b(2.)578 4755 y Fp(c\))j Fr(W)-8 b(e)33 b(need)h(to)e(sho)m(w)h(that)g(for)f(ev)m(ery)i Fo(t)2108 4770 y Fx(0)2148 4755 y Fr(,)1155 5020 y Fo(i)22 b Fr(lim)1205 5080 y Fm(\034)8 b Fy(!)p Fx(0)1377 4953 y Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)1589 4968 y Fx(0)1650 4953 y Fr(+)22 b Fo(\034)11 b Fr(\))22 b Fn(\000)h Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)2173 4968 y Fx(0)2212 4953 y Fr(\))p 1377 4997 874 4 v 1786 5089 a Fo(\034)2287 5020 y Fr(=)28 b Fo(H)8 b Fr(\()p Fo(t)2553 5035 y Fx(0)2592 5020 y Fr(\))p Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)2842 5035 y Fx(0)2881 5020 y Fr(\))p Fo(;)550 b Fr(\(61\))432 5268 y(where)45 b(the)g(limit)c(is)i(to)h(b)s(e)h(understo)s(o)s(d)g (in)e(the)i Fo(L)2428 5232 y Fx(2)2512 5268 y Fr(norm.)78 b(In)44 b(view)h(of)f(part)g(a\),)432 5388 y(p)s(oin)m(t)m(wise)30 b(con)m(v)m(ergence)k(for)c(ev)m(ery)j Fl(r)28 b Fn(2)g Fj(R)2036 5352 y Fx(3)2112 5388 y Fr(is)j(giv)m(en.)43 b(It)31 b(remains)f(to)g(sho)m(w)i(that)f(the)2002 5637 y(12)p eop %%Page: 13 13 13 12 bop 83 573 a Fr(left-hand)35 b(side)g(of)43 b(\(61\))35 b(is)g(b)s(ounded)h(b)m(y)h(a)e(commen)g Fo(L)2209 537 y Fx(2)2284 573 y Fr(function)h(for)f Fn(j)p Fo(\034)11 b Fn(j)33 b Fo(<)f Fr(1;)37 b(the)83 693 y(statemen)m(t)c(follo)m(ws)e (then)j(b)m(y)f(dominated)e(con)m(v)m(ergence.)47 b(F)-8 b(or)31 b Fo(\034)39 b(>)28 b Fr(0,)259 780 y Fk(\014)259 840 y(\014)259 900 y(\014)259 959 y(\014)292 780 y(\014)292 840 y(\014)292 900 y(\014)292 959 y(\014)336 857 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)566 872 y Fx(0)627 857 y Fr(+)22 b Fo(\034)11 b Fr(\))22 b Fn(\000)h Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)1168 872 y Fx(0)1207 857 y Fr(\))p 336 902 910 4 v 764 993 a Fo(\034)1255 780 y Fk(\014)1255 840 y(\014)1255 900 y(\014)1255 959 y(\014)1288 780 y(\014)1288 840 y(\014)1288 900 y(\014)1288 959 y(\014)1405 925 y Fr(=)130 b(sup)1565 1010 y Fm(\033)r Fy(2)p Fx(\(0)p Fm(;\034)8 b Fx(\))1820 840 y Fk(\014)1820 900 y(\014)1853 840 y(\014)1853 900 y(\014)1896 925 y Fo(@)1947 940 y Fm(t)1978 925 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fn(j)2273 954 y Fm(t)p Fx(=)p Fm(t)2378 963 y Fg(0)2413 954 y Fx(+)p Fm(\033)2515 840 y Fk(\014)2515 900 y(\014)2548 840 y(\014)2548 900 y(\014)1405 1145 y Fr(=)130 b(sup)1565 1231 y Fm(\033)r Fy(2)p Fx(\(0)p Fm(;\034)8 b Fx(\))1820 1145 y Fn(jj)o Fr(\()p Fo(H)g Fr(\()p Fo(t)2075 1160 y Fx(0)2137 1145 y Fr(+)22 b Fo(\033)t Fr(\))p Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b(t)2544 1160 y Fx(0)2605 1145 y Fr(+)22 b Fo(\033)t Fr(\)\))16 b(\()p Fl(r)p Fr(\))p Fn(j)o(j)1404 1437 y(\024)84 b Fo(C)69 b Fr(sup)1658 1523 y Fm(\033)r Fy(2)p Fx(\(0)p Fm(;\034)8 b Fx(\))2270 1369 y Fr(1)22 b(+)g Fn(j)p Fo(t)2502 1384 y Fx(0)2564 1369 y Fr(+)g Fo(\033)t Fn(j)2749 1333 y Fx(2)p 1924 1414 1212 4 v 1924 1505 a Fr(1)f(+)h Fo(x)2147 1476 y Fx(2)2209 1505 y Fr(+)g Fo(y)2359 1476 y Fx(2)2420 1505 y Fr(+)g(\()p Fo(z)27 b Fn(\000)c Fr(\()p Fo(t)2801 1520 y Fx(0)2862 1505 y Fr(+)f Fo(\033)t Fr(\)\))3095 1476 y Fx(2)3147 1657 y Fr(\(62\))83 1837 y(for)35 b(a)h(certain)f(constan)m (t)i Fo(C)7 b Fr(,)37 b(whic)m(h)f(is)g(clear)f(b)m(y)i(part)e(b\).)54 b(F)-8 b(or)35 b Fo(\034)45 b(<)33 b Fr(0,)j(w)m(e)h(obtain)e(a)83 1957 y(similar)29 b(result,)j(in)f(whic)m(h)h(sup)1239 1980 y Fm(\033)r Fy(2)p Fx(\(0)p Fm(;\034)8 b Fx(\))1513 1957 y Fr(is)32 b(replaced)g(b)m(y)h(sup)2276 1980 y Fm(\033)r Fy(2)p Fx(\()p Fm(\034)t(;)p Fx(0\))2515 1957 y Fr(.)43 b(With)32 b(little)d(e\013ort,)83 2077 y(w)m(e)34 b(see)f(that)g(for)f Fn(j)p Fo(\034)11 b Fn(j)27 b(\024)h Fr(1,)208 2184 y Fk(\014)208 2244 y(\014)208 2303 y(\014)208 2363 y(\014)241 2184 y(\014)241 2244 y(\014)241 2303 y(\014)241 2363 y(\014)284 2261 y Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)514 2276 y Fx(0)575 2261 y Fr(+)22 b Fo(\034)11 b Fr(\))23 b Fn(\000)f Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)1116 2276 y Fx(0)1155 2261 y Fr(\))p 284 2305 910 4 v 712 2397 a Fo(\034)1203 2184 y Fk(\014)1203 2244 y(\014)1203 2303 y(\014)1203 2363 y(\014)1237 2184 y(\014)1237 2244 y(\014)1237 2303 y(\014)1237 2363 y(\014)1297 2328 y Fn(\024)29 b Fo(C)1954 2261 y Fr(1)22 b(+)g(\()p Fn(j)p Fo(t)2224 2276 y Fx(0)2264 2261 y Fn(j)f Fr(+)h(1\))2498 2213 y Fx(2)p 1506 2305 1480 4 v 1506 2397 a Fr(1)g(+)g Fo(x)1730 2368 y Fx(2)1792 2397 y Fr(+)g Fo(y)1942 2368 y Fx(2)2003 2397 y Fr(+)g(max)o Fn(fj)p Fo(z)27 b Fn(\000)22 b Fo(t)2566 2412 y Fx(0)2606 2397 y Fn(j)g(\000)g Fr(1)p Fo(;)17 b Fr(0)p Fn(g)2947 2368 y Fx(2)2996 2328 y Fo(;)124 b Fr(\(63\))83 2569 y(where)28 b(the)g(righ)m(t-hand)e(side)i(is)e(an)h Fo(L)1475 2533 y Fx(2)1542 2569 y Fr(function)g(of)g Fo(x)p Fr(,)i Fo(y)t Fr(,)e(and)g Fo(z)t Fr(.)43 b(This)27 b(con\014rms)g(\(61\))83 2690 y(and)33 b(concludes)g(the)g(pro)s(of.)p 3250 2690 4 66 v 3254 2627 59 4 v 3254 2690 V 3312 2690 4 66 v 229 2873 a(F)-8 b(or)21 b(the)h(usual)g(F)-8 b(ourier)20 b(transformation,)i(the)g(Riemann-Leb)s(esgue)g(lemma)d(holds,)83 2994 y(whic)m(h)28 b(states)g(that)e(the)i(F)-8 b(ourier)26 b(transformation)e(de\014ned)29 b(on)e Fn(C)2506 2958 y Fy(1)2500 3018 y Fx(0)2608 2994 y Fr(extends)i(uniquely)83 3114 y(to)38 b(a)h(b)s(ounded)g(map)f(from)f Fo(L)1225 3078 y Fx(1)1304 3114 y Fr(in)m(to)g Fn(C)1559 3129 y Fy(1)1635 3114 y Fr(,)j(the)f(con)m(tin)m(uous)g(functions)g(v)-5 b(anishing)37 b(at)83 3235 y(in\014nit)m(y)-8 b(,)30 b(see)h(Theorem)g(IX.7)f(in.)1324 3198 y Fx(27)1441 3235 y Fr(In)g(the)h(case)g(of)f(the)g(V)-8 b(olk)m(o)m(v)31 b(solutions,)e(a)h(similar)83 3355 y(result)j(is)f(true:)83 3524 y Fl(Theorem)37 b(4.)188 3692 y Fp(a\))49 b(Ther)-5 b(e)34 b(is)h(a)f(c)-5 b(onstant)35 b Fo(C)42 b Fp(with)1287 3872 y Fn(jj)o(F)1414 3887 y Fm(t)1444 3872 y Fr(\()p Fo(\036)p Fr(\))p Fn(j)o(j)1633 3901 y Fy(1)1735 3872 y Fn(\024)28 b Fo(C)c Fn(j)o(j)p Fo(\036)p Fn(j)o(j)2102 3901 y Fm(L)2150 3882 y Fg(1)2185 3901 y Fx(\()p Fe(R)2260 3882 y Fg(3)2295 3901 y Fx(\))2322 3882 y Fg(4)3147 3872 y Fr(\(64\))327 4064 y Fp(for)35 b(al)5 b(l)34 b Fo(\036)27 b Fn(2)i(C)860 4028 y Fy(1)854 4089 y Fx(0)935 4064 y Fr(\()p Fj(R)1038 4028 y Fx(3)1084 4064 y Fr(\))1122 4028 y Fx(4)1189 4064 y Fn(\032)f 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Fx(3)2753 5388 y Fr(\))2791 5352 y Fx(4)2830 5388 y Fp(.)1653 5637 y Fr(13)p eop %%Page: 14 14 14 13 bop 432 573 a Fp(Pr)-5 b(o)g(of.)48 b(a\))42 b Fr(In)e(order)f(to)f(adapt)h(the)g(pro)s(of)g(for)f(the)h(usual)g(F)-8 b(ourier)38 b(transformation)432 693 y(in,)541 657 y Fx(27)643 693 y Fr(w)m(e)29 b(ha)m(v)m(e)g(to)e(c)m(hec)m(k)j(\014rst)e (that)f(the)h(V)-8 b(olk)m(o)m(v)28 b(solutions)f(are)g(b)s(ounded)i (in)d Fl(p)p Fr(.)42 b(Since)432 814 y(this)29 b(is)h(not)g(directly)f (clear)h(from)f(the)h(de\014ning)g(equation)g(\(9\),)g(w)m(e)h(shall)e (\014rst)i(rewrite)432 934 y(the)i(V)-8 b(olk)m(o)m(v)32 b(solutions.)43 b(T)-8 b(o)33 b(this)f(end,)h(observ)m(e)h(that)1683 1171 y Fo(=)1678 1168 y(n)1748 1127 y Fr(~)1756 1159 y Fo(=)1736 1168 y(A)p Fr(\()p Fo(\030)5 b Fr(\))27 b(=)h Fn(\000)2153 1127 y Fr(~)2161 1159 y Fo(=)2141 1168 y(A)q Fr(\()p Fo(\030)5 b Fr(\))2344 1171 y Fo(=)2339 1168 y(n)o(:)1073 b Fr(\(67\))432 1388 y(Emplo)m(ying)31 b(\(17\),)h(w)m(e)h (can)g(write)1162 1622 y Fo( )1225 1637 y Fi(p)p Fm(;r)1328 1622 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(=)1685 1598 y Fl(~)1660 1622 y(M)1766 1581 y Fx(\()p Fm(\017)1822 1589 y Fh(r)1857 1581 y Fx(\))1888 1622 y Fr(\()p Fl(p)p Fo(;)17 b(t)22 b Fn(\000)h Fo(z)t Fr(\))p Fo(u)2332 1637 y Fm(r)2370 1622 y Fr(\()p Fl(p)p Fr(\))p Fo(e)2553 1581 y Fm(iS)2624 1558 y Fg(\()p Fh(\017)2674 1566 y(r)2709 1558 y Fg(\))2737 1581 y Fx(\()p Fi(p)p Fm(;)p Fi(r)p Fm(;t)p Fx(\))3496 1622 y Fr(\(68\))432 1842 y(with)754 2077 y Fl(~)729 2101 y(M)835 2060 y Fx(\()p Fy(\006)p Fx(\))949 2101 y Fr(\()p Fl(p)p Fo(;)17 b(\030)5 b Fr(\))82 b(:=)h Fl(1)22 b Fn(\007)h Fo(e)1682 2060 y Fr(~)1690 2092 y Fo(=)1670 2101 y(A)p Fr(\()p Fo(\030)5 b Fr(\))1951 2024 y Fo(E)h Fr(\()p Fl(p)p Fr(\))2172 2027 y Fo(=)2167 2024 y(n)22 b Fn(\006)h Fo(m)f Fn(\000)2554 2037 y Fo(=)2554 2024 y(p)p 1893 2078 768 4 v 1893 2169 a Fr(2)p Fo(E)6 b Fr(\()p Fl(p)p Fr(\)\()p Fo(E)g Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(p)2583 2184 y Fm(z)2623 2169 y Fr(\))1275 2374 y(=)96 b Fl(1)22 b Fn(\007)h Fo(e)1682 2333 y Fr(~)1690 2365 y Fo(=)1670 2374 y(A)p Fr(\()p Fo(\030)5 b Fr(\))1884 2233 y Fk(\022)1966 2306 y Fn(\006)p Fo(m)23 b Fr(+)f Fo(p)2298 2321 y Fm(x)2342 2306 y Fo(\015)2398 2270 y Fx(1)2460 2306 y Fr(+)g Fo(p)2607 2321 y Fm(y)2648 2306 y Fo(\015)2704 2270 y Fx(2)p 1966 2351 778 4 v 1971 2442 a Fr(2)p Fo(E)6 b Fr(\()p Fl(p)p Fr(\)\()p Fo(E)g Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(p)2661 2457 y Fm(z)2701 2442 y Fr(\))2776 2374 y Fn(\000)2993 2306 y Fr(1)p 2885 2351 265 4 v 2885 2442 a(2)p Fo(E)6 b Fr(\()p Fl(p)p Fr(\))3160 2374 y Fo(\015)3216 2333 y Fx(3)3256 2233 y Fk(\023)3346 2374 y Fo(:)123 b Fr(\(69\))432 2645 y(Using)32 b(the)h(equalit)m(y)826 2847 y(1)p 491 2891 719 4 v 491 2983 a Fo(E)6 b Fr(\()p Fl(p)p Fr(\)\()p Fo(E)g Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(p)1132 2998 y Fm(z)1172 2983 y Fr(\))1242 2914 y(+)1684 2847 y(1)p 1350 2891 718 4 v 1350 2983 a Fo(E)6 b Fr(\()p Fl(p)p Fr(\)\()p Fo(E)g Fr(\()p Fl(p)p Fr(\))22 b(+)g Fo(p)1989 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a(\(2)p Fo(\031)t Fr(\))1838 4523 y Fx(3)p Fm(=)p Fx(2)2029 4359 y(4)1974 4389 y Fk(X)1984 4598 y Fm(r)r Fx(=1)2135 4348 y Fk(Z)2190 4574 y Fe(R)2238 4555 y Fg(3)2294 4484 y Fn(j)o Fo(\036)2379 4499 y Fm(r)2417 4484 y Fr(\()p Fl(p)p Fr(\))p Fn(j)16 b(jj)p Fo( )2718 4499 y Fi(p)p Fm(;r)2821 4484 y Fr(\()p Fl(r)p Fo(;)h(t)p Fr(\))p Fn(j)o(j)44 b Fo(d)3172 4442 y Fx(3)3211 4484 y Fo(p)1484 4819 y Fn(\024)1777 4751 y Fr(1)p 1654 4796 V 1654 4887 a(\(2)p Fo(\031)t Fr(\))1838 4858 y Fx(3)p Fm(=)p Fx(2)2029 4694 y(4)1974 4724 y Fk(X)1984 4933 y Fm(r)r Fx(=1)2135 4819 y Fo(c)2177 4834 y Fm(r)2231 4819 y Fn(jj)p Fo(\036)2345 4834 y Fm(r)2382 4819 y Fn(jj)2438 4848 y Fm(L)2486 4829 y Fg(1)2541 4819 y Fo(:)928 b Fr(\(73\))578 5125 y Fp(b\))46 b Fr(This)d(follo)m(ws)f(as)h(for)f(the)h(Riemann-Leb) s(esgue)g(lemma:)62 b(F)-8 b(or)42 b(a)g(giv)m(en)h Fo(\036)i Fn(2)432 5246 y Fo(L)498 5210 y Fx(1)538 5246 y Fr(\()p Fj(R)641 5210 y Fx(3)687 5246 y Fr(\))725 5210 y Fx(4)764 5246 y Fr(,)e(c)m(ho)s(ose)e(a)f(sequence)j(\()p Fo(\036)1748 5261 y Fm(n)1795 5246 y Fr(\))1833 5275 y Fm(n)p Fy(2)p Fe(N)2015 5246 y Fr(in)c Fn(C)2194 5210 y Fy(1)2188 5270 y Fx(0)2269 5246 y Fr(\()p Fj(R)2373 5210 y Fx(3)2419 5246 y Fr(\))2457 5210 y Fx(4)2536 5246 y Fr(with)h Fo(\036)2824 5261 y Fm(n)2912 5246 y Fn(!)h Fo(\036)p Fr(,)h Fo(n)f Fn(!)f(1)p Fr(,)i(in)432 5388 y(the)j Fo(L)678 5352 y Fx(1)718 5388 y Fr(\()p Fj(R)822 5352 y Fx(3)867 5388 y Fr(\))905 5352 y Fx(4)989 5388 y Fr(norm.)80 b(By)45 b(Theorem)g(2,)j Fn(F)2114 5337 y Fx(\(1\))2104 5411 y Fm(t)2208 5388 y Fr(\()p Fo(\036)2304 5403 y Fm(n)2350 5388 y Fr(\))h(=)f Fn(F)2633 5403 y Fm(t)2663 5388 y Fr(\()p Fo(\036)2759 5403 y Fm(n)2805 5388 y Fr(\))h Fn(2)g(C)3059 5403 y Fy(1)3134 5388 y Fr(\()p Fj(R)3238 5352 y Fx(3)3283 5388 y Fr(\))3321 5352 y Fx(4)3405 5388 y Fr(for)c(all)2002 5637 y(14)p eop %%Page: 15 15 15 14 bop 83 573 a Fo(n)48 b Fn(2)g Fj(N)9 b Fr(.)85 b(On)45 b(the)g(other)f(hand,)k(part)c(a\))g(of)g(the)h(presen)m(t)h (theorem)f(ensures)h(that)83 707 y Fn(F)165 656 y Fx(\(1\))155 729 y Fm(t)259 707 y Fr(\()p Fo(\036)355 722 y Fm(n)401 707 y Fr(\))k Fn(!)f(F)720 656 y Fx(\(1\))710 729 y Fm(t)813 707 y Fr(\()p Fo(\036)p Fr(\),)g Fo(n)g Fn(!)g(1)p Fr(,)f(uniformly)43 b(on)i Fj(R)2125 671 y Fx(3)2171 707 y Fr(.)81 b(Therefore,)50 b(w)m(e)c(also)e(ha)m(v)m(e)83 827 y Fn(F)155 842 y Fm(t)184 827 y Fr(\()p Fo(\036)p Fr(\))28 b Fn(2)g(C)492 842 y Fy(1)567 827 y Fr(\()p Fj(R)671 791 y Fx(3)716 827 y Fr(\))754 791 y Fx(4)794 827 y Fr(.)229 948 y Fp(c\))52 b Fr(In)d(view)g(of)55 b(\(72\))o(,)d(the)d(in)m(tegral)e(on)h(the)h (righ)m(t-hand)f(side)g(of)55 b(\(66\))48 b(exists)83 1068 y(for)41 b Fo(\036)i Fn(2)h Fo(L)518 1032 y Fx(1)557 1068 y Fr(\()p Fj(R)661 1032 y Fx(3)707 1068 y Fr(\))745 1032 y Fx(4)784 1068 y Fr(.)71 b(T)-8 b(aking)41 b(again)f(a)i (sequence)i(as)e(in)f(the)h(pro)s(of)f(of)g(part)g(b\),)j(the)83 1188 y(statemen)m(t)33 b(follo)m(ws)e(b)m(y)j(dominated)d(con)m(v)m (ergence.)p 3250 1188 4 66 v 3254 1126 59 4 v 3254 1188 V 3312 1188 4 66 v 83 1513 a Fq(V.)54 b(\\Normalization")83 1732 y Fr(The)31 b(follo)m(wing)c(theorem)i(ma)m(y)h(b)s(e)g (considered)g(as)g(the)g(cen)m(tral)g(p)s(oin)m(t)f(of)g(this)g (article:)83 1888 y Fl(Theorem)47 b(5.)f Fp(L)-5 b(et)891 1864 y Fl(~)877 1888 y(A)42 b Fn(2)h(C)1171 1852 y Fx(1)1165 1914 y Fm(b)1210 1888 y Fp(.)69 b(Then)42 b(for)g(every)h Fo(t)f Fn(2)h Fj(R)5 b Fp(,)51 b Fn(F)2405 1903 y Fm(t)2477 1888 y Fp(is)42 b(an)g(isometry)h(with)83 2008 y(r)-5 b(esp)g(e)g(ct)35 b(to)g(the)g Fo(L)746 1972 y Fx(2)820 2008 y Fp(norm,)f(i.)h(e.)45 b(for)34 b(every)h Fo(\036)27 b Fn(2)i Fo(L)1984 1972 y Fx(2)2023 2008 y Fr(\()p Fj(R)2127 1972 y Fx(3)2173 2008 y Fr(\))2211 1972 y Fx(4)2250 2008 y Fp(,)1099 2173 y Fn(j)o(jF)1226 2188 y Fm(t)1255 2173 y Fr(\()p Fo(\036)p Fr(\))p Fn(jj)1444 2202 y Fm(L)1492 2183 y Fg(2)1527 2202 y Fx(\()p Fe(R)1602 2183 y Fg(3)1637 2202 y Fx(\))1664 2183 y Fg(4)1731 2173 y Fr(=)e Fn(jj)p Fo(\036)p Fn(j)o(j)2003 2202 y Fm(L)2051 2183 y Fg(2)2085 2202 y Fx(\()p Fe(R)2160 2183 y Fg(3)2195 2202 y Fx(\))2222 2183 y Fg(4)2278 2173 y Fo(:)842 b Fr(\(74\))229 2337 y(As)33 b(a)g(simple)e(conclusion,)h(w)m(e)i(note:)83 2493 y Fl(Corollary)j(6.)42 b Fp(Under)35 b(the)g(assumptions)f(of)g (The)-5 b(or)g(em)34 b(5,)960 2657 y Fn(hF)1071 2672 y Fm(t)1100 2657 y Fr(\()p Fo(\036)p Fr(\))p Fo(;)17 b Fn(F)1350 2672 y Fm(t)1379 2657 y Fr(\()p Fo(\037)p Fr(\))p Fn(i)1555 2687 y Fm(L)1603 2668 y Fg(2)1637 2687 y Fx(\()p Fe(R)1712 2668 y Fg(3)1747 2687 y Fx(\))1774 2668 y Fg(4)1841 2657 y Fr(=)28 b Fn(h)o Fo(\036;)17 b(\037)p Fn(i)2185 2687 y Fm(L)2233 2668 y Fg(2)2267 2687 y Fx(\()p Fe(R)2342 2668 y Fg(3)2377 2687 y Fx(\))2404 2668 y Fg(4)3147 2657 y Fr(\(75\))83 2834 y Fp(for)35 b(every)f Fo(\036;)17 b(\037)28 b Fn(2)g Fo(L)847 2798 y Fx(2)887 2834 y Fr(\()p Fj(R)990 2798 y Fx(3)1036 2834 y Fr(\))1074 2798 y Fx(4)1113 2834 y Fp(.)229 2990 y Fr(The)40 b(length)m(y)f(pro)s(of)e(of)h(Theorem)h(5)g(will)d(b)s(e)j (deferred)g(to)f(App)s(endix)i(A.)61 b(Here)83 3111 y(w)m(e)49 b(shall)d(only)i(v)m(erify)g(Corollary)e(6)i(and)g(then)g(p)s(oin)m(t)f (out)h(the)g(connection)g(with)83 3231 y(the)38 b(\\normalization)c(to) k(a)g(delta)f(function")g(and)h(\\orthogonalit)m(y")e(of)h(the)i(V)-8 b(olk)m(o)m(w)83 3351 y(solutions.)83 3531 y Fp(Pr)j(o)g(of)35 b(of)f(Cor)-5 b(ol)5 b(lary)35 b(6.)48 b Fr(This)29 b(corollary)e(is)h (just)h(the)g(application)d(of)j(a)f(general)g(theo-)83 3651 y(rem)j(v)-5 b(alid)30 b(for)g(an)m(y)j(isometry)d(b)s(et)m(w)m (een)k(t)m(w)m(o)e(complex)f(Hilb)s(ert)f(spaces:)44 b(W)-8 b(e)32 b(use)g(the)83 3771 y(p)s(olarization)d(iden)m(tit)m(y) 306 3975 y Fn(h)p Fo(x;)17 b(y)t Fn(i)27 b Fr(=)675 3907 y(1)p 675 3952 49 4 v 675 4043 a(4)750 3894 y Fk(\000)796 3975 y Fn(jj)o Fo(x)c Fr(+)f Fo(y)t Fn(j)o(j)1133 3926 y Fx(2)1195 3975 y Fr(+)g Fn(jj)o Fo(x)h Fn(\000)g Fo(y)t Fn(j)n(j)1632 3926 y Fx(2)1694 3975 y Fr(+)f Fo(i)17 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Fo(T)46 b Fr(\(satisfying)32 b Fn(jj)o Fo(T)14 b(z)t Fn(j)q(j)27 b Fr(=)g Fn(jj)p Fo(z)t Fn(jj)32 b Fr(for)g(all)f Fo(z)t Fr(\),)i(this)f(leads)h(to)172 4779 y Fn(h)o Fo(T)14 b(x;)j(T)d(y)t Fn(i)255 4968 y Fr(=)424 4901 y(1)p 424 4945 V 424 5037 a(4)499 4888 y Fk(\000)545 4968 y Fn(j)o(j)p Fo(T)g Fr(\()p Fo(x)22 b Fr(+)g Fo(y)t Fr(\))p Fn(j)o(j)1029 4920 y Fx(2)1090 4968 y Fr(+)g Fn(jj)o Fo(T)14 b Fr(\()p Fo(x)23 b Fn(\000)f Fo(y)t Fr(\))p Fn(j)o(j)1674 4920 y Fx(2)1735 4968 y Fr(+)g Fo(i)17 b Fn(jj)p Fo(T)d Fr(\()p Fo(x)22 b Fn(\000)g Fo(iy)t Fr(\))p Fn(j)o(j)2402 4920 y Fx(2)2464 4968 y Fn(\000)g Fo(i)17 b Fn(jj)p Fo(T)d Fr(\()p Fo(x)22 b Fr(+)g Fo(iy)t Fr(\))p Fn(j)o(j)3131 4920 y Fx(2)3170 4888 y Fk(\001)255 5202 y Fr(=)424 5134 y(1)p 424 5179 V 424 5270 a(4)499 5121 y Fk(\000)545 5202 y Fn(j)o(j)p Fo(x)g Fr(+)g Fo(y)t Fn(j)o(j)882 5153 y Fx(2)944 5202 y Fr(+)g Fn(j)o(j)p Fo(x)g Fn(\000)h Fo(y)t Fn(j)o(j)1381 5153 y Fx(2)1442 5202 y Fr(+)f Fo(i)17 b Fn(jj)o Fo(x)23 b Fn(\000)g Fo(iy)t Fn(j)o(j)1962 5153 y Fx(2)2024 5202 y Fn(\000)g Fo(i)17 b Fn(j)o(j)p Fo(x)22 b Fr(+)g Fo(iy)t Fn(j)o(j)2544 5153 y Fx(2)2584 5121 y Fk(\001)2657 5202 y Fr(=)27 b Fn(h)p Fo(x;)17 b(y)t Fn(i)f Fo(:)115 b Fr(\(78\))p 3250 5388 4 66 v 3254 5326 59 4 v 3254 5388 V 3312 5388 4 66 v 1653 5637 a(15)p eop %%Page: 16 16 16 15 bop 578 573 a Fr(In)29 b(the)g(ph)m(ysical)f(literature,)g(e.)h (g.,)1884 537 y Fx(7,)9 b(13,)g(23,)g(28)2251 573 y Fr(the)29 b(\\normalization")24 b(and)29 b(\\orthog-)432 693 y(onalit)m(y")i (prop)s(ert)m(y)i(is)f(laxly)g(written)g(as)h(follo)m(ws:)995 813 y Fk(Z)1050 1039 y Fe(R)1098 1020 y Fg(3)1154 949 y Fo( )1221 908 y Fy(y)1217 973 y Fi(p)p Fm(;r)1320 949 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fo( )1584 964 y Fi(p)1629 945 y Fc(0)1651 964 y Fm(;r)1705 945 y Fc(0)1731 949 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))g Fo(d)2000 908 y Fx(3)2039 949 y Fo(r)30 b Fr(=)d(\(2)p Fo(\031)t Fr(\))2400 908 y Fx(3)2439 949 y Fo(\016)2482 964 y Fm(r)n(;r)2566 945 y Fc(0)2593 949 y Fo(\016)2640 908 y Fx(\(3\))2734 949 y Fr(\()p Fl(p)22 b Fn(\000)h Fl(p)3018 908 y Fy(0)3041 949 y Fr(\))p Fo(;)390 b Fr(\(79\))432 1225 y(where)31 b Fo(\016)758 1189 y Fx(\(3\))882 1225 y Fr(is)e(the)h (three-dimensional)e(delta)h(function.)42 b(The)31 b(prop)s(ert)m(y)h (\(79\))d(is)g(used)432 1345 y(for)j(the)h(v)m(eri\014cation)f(of)39 b(\(75\))o(:)706 1648 y Fn(h)o(F)816 1663 y Fm(t)846 1648 y Fr(\()p Fo(\036)p Fr(\))p Fo(;)17 b Fn(F)1096 1663 y Fm(t)1124 1648 y Fr(\()p Fo(\037)p Fr(\))p Fn(i)83 b Fr(=)1639 1581 y(1)p 1552 1625 223 4 v 1552 1716 a(\(2)p Fo(\031)t Fr(\))1736 1688 y Fx(3)1882 1524 y(4)1827 1553 y Fk(X)1801 1766 y Fm(r)n(;r)1885 1747 y Fc(0)1907 1766 y Fx(=1)2014 1512 y Fk(Z)2069 1738 y Fe(R)2117 1719 y Fg(3)2172 1508 y Fk(\022)2262 1512 y(Z)2318 1738 y Fe(R)2366 1719 y Fg(3)2421 1512 y Fk(Z)2476 1738 y Fe(R)2524 1719 y Fg(3)p 2580 1561 234 4 v 2580 1648 a Fo(\036)2638 1663 y Fm(r)2675 1648 y Fr(\()p Fl(p)p Fr(\))p Fo(\037)2874 1663 y Fm(r)2908 1644 y Fc(0)2935 1648 y Fr(\()p Fl(p)3035 1607 y Fy(0)3058 1648 y Fr(\))1542 1967 y Fn(\002)17 b(h)p Fo( )1738 1982 y Fi(p)p Fm(;r)1841 1967 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))p Fo(;)g( )2149 1982 y Fi(p)2194 1963 y Fc(0)2216 1982 y Fm(r)2250 1963 y Fc(0)2276 1967 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))p Fn(i)2516 1999 y Fe(C)2558 1980 y Fg(4)2636 1967 y Fo(d)2687 1925 y Fx(3)2726 1967 y Fo(p)g(d)2843 1925 y Fx(3)2881 1967 y Fo(p)2930 1925 y Fy(0)2954 1826 y Fk(\023)3027 1967 y Fo(d)3078 1925 y Fx(3)3117 1967 y Fo(r)1383 2272 y Fr(=)1639 2204 y(1)p 1552 2249 223 4 v 1552 2340 a(\(2)p Fo(\031)t Fr(\))1736 2311 y Fx(3)1882 2147 y(4)1827 2177 y Fk(X)1801 2390 y Fm(r)n(;r)1885 2371 y Fc(0)1907 2390 y Fx(=1)2014 2136 y Fk(Z)2069 2362 y Fe(R)2117 2343 y Fg(3)2172 2136 y Fk(Z)2228 2362 y Fe(R)2276 2343 y Fg(3)p 2331 2185 234 4 v 2331 2272 a Fo(\036)2389 2287 y Fm(r)2427 2272 y Fr(\()p Fl(p)p Fr(\))p Fo(\037)2626 2287 y Fm(r)2660 2268 y Fc(0)2686 2272 y Fr(\()p Fl(p)2786 2231 y Fy(0)2810 2272 y Fr(\))1542 2590 y Fn(\002)1619 2450 y Fk(\022)1709 2455 y(Z)1764 2680 y Fe(R)1812 2661 y Fg(3)1868 2590 y Fn(h)p Fo( )1970 2605 y Fi(p)p Fm(;r)2073 2590 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))p Fo(;)g( )2381 2605 y Fi(p)2426 2586 y Fc(0)2448 2605 y Fm(r)2482 2586 y Fc(0)2508 2590 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))p Fn(i)2748 2622 y Fe(C)2790 2603 y Fg(4)2868 2590 y Fo(d)2919 2549 y Fx(3)2958 2590 y Fo(r)3005 2450 y Fk(\023)3078 2590 y Fo(d)3129 2549 y Fx(3)3168 2590 y Fo(p)g(d)3285 2549 y Fx(3)3324 2590 y Fo(p)3373 2549 y Fy(0)1383 2895 y Fr(=)1596 2771 y Fx(4)1542 2801 y Fk(X)1552 3010 y Fm(r)r Fx(=1)1702 2760 y Fk(Z)1758 2985 y Fe(R)1806 2966 y Fg(3)1861 2760 y Fk(Z)1916 2985 y Fe(R)1964 2966 y Fg(3)p 2020 2809 V 2020 2895 a Fo(\036)2078 2910 y Fm(r)2115 2895 y Fr(\()p Fl(p)p Fr(\))p Fo(\037)2314 2910 y Fm(r)2352 2895 y Fr(\()p Fl(p)2452 2854 y Fy(0)2476 2895 y Fr(\))p Fo(\016)2561 2854 y Fx(\(3\))2655 2895 y Fr(\()p Fl(p)22 b Fn(\000)h Fl(p)2939 2854 y Fy(0)2963 2895 y Fr(\))17 b Fo(d)3069 2854 y Fx(3)3107 2895 y Fo(p)g(d)3224 2854 y Fx(3)3263 2895 y Fo(p)3312 2854 y Fy(0)1383 3230 y Fr(=)1596 3106 y Fx(4)1542 3136 y Fk(X)1552 3345 y Fm(r)r Fx(=1)1702 3095 y Fk(Z)1758 3320 y Fe(R)1806 3301 y Fg(3)p 1861 3144 V 1861 3230 a Fo(\036)1919 3245 y Fm(r)1957 3230 y Fr(\()p Fl(p)p Fr(\))p Fo(\037)2156 3245 y Fm(r)2194 3230 y Fr(\()p Fl(p)p Fr(\))g Fo(d)2400 3189 y Fx(3)2439 3230 y Fo(p)1383 3463 y Fr(=)83 b Fn(h)o Fo(\036;)17 b(\037)p Fn(i)1782 3492 y Fm(L)1830 3473 y Fg(2)1864 3492 y Fx(\()p Fe(R)1939 3473 y Fg(3)1974 3492 y Fx(\))2001 3473 y Fg(4)2057 3463 y Fo(:)1412 b Fr(\(80\))432 3675 y(This)38 b(calculation)d(certainly)i(in)m(v)m(olv)m(es)h(an)g (unjusti\014ed)g(in)m(terc)m(hange)h(of)e(in)m(tegration)432 3796 y(orders)47 b(and)f(is)g(therefore)g(rather)h(questionable)f(from) f(a)h(mathematical)d(p)s(oin)m(t)i(of)432 3916 y(view.)578 4037 y(W)-8 b(e)22 b(men)m(tion)f(that)g(a)h(di\013eren)m(t)g (\\normalization")17 b(has)23 b(b)s(een)f(emplo)m(y)m(ed)g(b)m(y)h (Neville)432 4157 y(and)34 b(Rohrlic)m(h)986 4121 y Fx(17)1094 4157 y Fr(and)g(Bergou)g(and)h(V)-8 b(arr\023)-49 b(o.)2079 4121 y Fx(6)2166 4157 y Fr(These)36 b(authors)f(use)g(a)f(scalar)f(pro) s(duct)432 4277 y(that)j(is)g(calculated)f(on)h(those)h(h)m(yp)s (erplanes)h(in)e(Mink)m(o)m(wski)h(space-time)f(where)h(the)432 4398 y(v)m(ector)f(p)s(oten)m(tial)d Fl(A)h Fr(is)g(constan)m(t,)i(i.)e (e.)51 b(their)34 b(considerations)g(in)m(v)m(olv)m(e)h(in)m(tegrals)f (of)432 4518 y(the)f(form)1536 4526 y Fk(Z)1591 4752 y Fy(f)p Fx(\()p Fi(r)p Fm(;t)p Fx(\))p Fy(2)p Fe(R)1854 4733 y Fg(4)1889 4752 y Fx(:)p Fm(t)p Fy(\000)p Fm(z)s Fx(=)p Fm(u)p Fy(g)2177 4662 y Fn(\001)17 b(\001)g(\001)31 b Fo(d)2377 4621 y Fx(3)2416 4662 y Fo(r)20 b(dt)930 b Fr(\(81\))432 4907 y(for)39 b(\014xed)j Fo(u)e Fn(2)h Fj(R)5 b Fr(.)73 b(Ho)m(w)m(ev)m(er,)45 b(the)40 b Fo(L)1872 4870 y Fx(2)1952 4907 y Fr(norms)g(determined)g(from)f(this)h(in)m (tegration)432 5027 y(cannot)i(b)s(e)h(regarded)f(as)h(a)f (normalization)c(for)k(the)g(probabilit)m(y)f(to)g(\014nd)i(a)f(parti-) 432 5147 y(cle.)h(F)-8 b(or)31 b(a)i(probabilit)m(y)d(in)m (terpretation,)h(one)i(w)m(ould)f(ha)m(v)m(e)i(to)e(in)m(tegrate)g(o)m (v)m(er)h(space)432 5268 y(for)j(a)g(\014xed)h(time,)f(at)g(least)g (for)g(a)g(certain)g(sp)s(eci\014ed)i(inertial)33 b(frame.)55 b(That)36 b(is,)h(one)432 5388 y(w)m(ould)30 b(need)g(a)g(space-lik)m (e)g(h)m(yp)s(erplane,)h(whereas)h(the)e(in)m(tegration)e(in)h(\(81\))g (is)h(o)m(v)m(er)h(a)2002 5637 y(16)p eop %%Page: 17 17 17 16 bop 83 573 a Fr(ligh)m(t-lik)m(e)29 b(h)m(yp)s(erplane.)44 b(Ev)m(en)34 b(though)e(the)h(ligh)m(t-lik)m(e)c(h)m(yp)s(erplane)k (and)f(the)g(related)83 693 y(co)s(ordinates)42 b(seem)g(more)f (adopted)i(to)e(the)i(symmetry)f(of)g(the)g(problem,)i(an)d(in)m(te-)83 814 y(gration)c(o)m(v)m(er)j(a)e(space-lik)m(e)h(h)m(yp)s(erplane)h(as) f(in)f(\(75\))g(or)g(\(79\))g(app)s(ears)h(m)m(uc)m(h)h(more)83 934 y(\\ph)m(ysical".)83 1280 y Fq(VI.)54 b(Extension)e(to)i Fb(L)1407 1228 y Fr(2)83 1504 y(Theorem)33 b(2)f(allo)m(ws)g(us)h(to)f (pronounce)i(the)f(follo)m(wing)c(de\014nition:)83 1718 y Fl(De\014nition)36 b(2.)42 b Fp(In)34 b(the)h(c)-5 b(ase)1247 1694 y Fl(~)1233 1718 y(A)27 b Fn(2)i(C)1498 1682 y Fx(1)1492 1744 y Fm(b)1537 1718 y Fp(,)35 b(let)1163 1959 y Fn(F)1245 1909 y Fx(\(2\))1235 1982 y Fm(t)1366 1959 y Fr(:)28 b Fo(L)1487 1918 y Fx(2)1527 1959 y Fr(\()p Fj(R)1631 1918 y Fx(3)1676 1959 y Fr(\))1714 1918 y Fx(4)1781 1959 y Fn(!)g Fo(L)1975 1918 y Fx(2)2014 1959 y Fr(\()p Fj(R)2118 1918 y Fx(3)2164 1959 y Fr(\))2202 1918 y Fx(4)3147 1959 y Fr(\(82\))83 2186 y Fp(b)-5 b(e)35 b(the)f(unique)h(isometric)f (extension)g(of)h Fn(F)1723 2201 y Fm(t)1787 2186 y Fp(fr)-5 b(om)34 b(Def.)g(1)h(to)g Fo(L)2498 2150 y Fx(2)2538 2186 y Fr(\()p Fj(R)2642 2150 y Fx(3)2687 2186 y Fr(\))2725 2150 y Fx(4)2765 2186 y Fp(.)229 2414 y Fr(It)30 b(will)e(b)s(e)i (pro\014table)f(to)h(ha)m(v)m(e)h(an)e(in)m(tegral)g(represen)m(tation) h(of)g Fn(F)2729 2363 y Fx(\(2\))2719 2436 y Fm(t)2852 2414 y Fr(for)f(a)h(larger)83 2534 y(subspace)h(of)e Fo(L)661 2498 y Fx(2)730 2534 y Fr(than)g(just)h Fn(C)1202 2498 y Fy(1)1196 2559 y Fx(0)1277 2534 y Fr(.)42 b(This)30 b(is)f(indeed)g(p)s(ossible)g(for)g(amplitude)e(functions)83 2655 y(in)32 b Fo(L)263 2619 y Fx(1)325 2655 y Fn(\\)22 b Fo(L)479 2619 y Fx(2)519 2655 y Fr(:)83 2869 y Fl(Theorem)37 b(7.)42 b Fp(F)-7 b(or)34 b Fo(\036)27 b Fn(2)h Fo(L)1114 2833 y Fx(1)1154 2869 y Fr(\()p Fj(R)1258 2833 y Fx(3)1303 2869 y Fr(\))1341 2833 y Fx(4)1403 2869 y Fn(\\)22 b Fo(L)1557 2833 y Fx(2)1597 2869 y Fr(\()p Fj(R)1701 2833 y Fx(3)1747 2869 y Fr(\))1785 2833 y Fx(4)1824 2869 y Fp(,)35 b(we)f(have)1076 3109 y Fn(F)1158 3058 y Fx(\(2\))1148 3132 y Fm(t)1251 3109 y Fr(\()p Fo(\036)p Fr(\))28 b(=)f Fn(F)1598 3058 y Fx(\(1\))1588 3132 y Fm(t)1692 3109 y Fr(\()p Fo(\036)p Fr(\))g Fn(2)h(C)1999 3124 y Fy(1)2075 3109 y Fr(\()p Fj(R)2178 3068 y Fx(3)2224 3109 y Fr(\))2262 3068 y Fx(4)2301 3109 y Fo(;)819 b Fr(\(83\))83 3336 y Fp(or)35 b(mor)-5 b(e)34 b(explicitly,)637 3537 y Fk(\020)696 3647 y Fn(F)778 3596 y Fx(\(2\))768 3670 y Fm(t)872 3647 y Fr(\()p Fo(\036)p Fr(\))1006 3537 y Fk(\021)1082 3647 y Fr(\()p Fl(r)p Fr(\))27 b(=)1467 3580 y(1)p 1345 3624 294 4 v 1345 3715 a(\(2)p Fo(\031)t Fr(\))1529 3687 y Fx(3)p Fm(=)p Fx(2)1719 3523 y(4)1665 3553 y Fk(X)1675 3762 y Fm(r)r Fx(=1)1825 3512 y Fk(Z)1881 3737 y Fe(R)1929 3718 y Fg(3)1984 3647 y Fo(\036)2042 3662 y Fm(r)2080 3647 y Fr(\()p Fl(p)p Fr(\))p Fo( )2281 3662 y Fi(p)p Fm(;r)2384 3647 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)2653 3606 y Fx(3)2691 3647 y Fo(p:)380 b Fr(\(84\))83 3958 y Fp(Pr)-5 b(o)g(of.)48 b Fr(W)-8 b(e)35 b(adapt)e(here)i(a)e(pro) s(of)g(from)f(the)i(usual)g(F)-8 b(ourier)32 b(transformation,)g(whic)m (h)83 4078 y(can)25 b(b)s(e)f(found)h(in)f(the)g(b)s(o)s(ok)g(of)g(W)-8 b(eidmann.)1728 4042 y Fx(29)1843 4078 y Fr(Let)25 b(\()p Fo(\036)2106 4093 y Fm(n)2152 4078 y Fr(\))2190 4107 y Fm(n)2262 4078 y Fr(b)s(e)f(a)g(sequence)k(in)23 b Fn(C)3019 4042 y Fy(1)3013 4103 y Fx(0)3094 4078 y Fr(\()p Fj(R)3198 4042 y Fx(3)3244 4078 y Fr(\))3282 4042 y Fx(4)83 4198 y Fr(with)33 b Fo(\036)364 4213 y Fm(n)440 4198 y Fn(!)c Fo(\036)p Fr(,)34 b Fo(n)c Fn(!)f(1)p Fr(,)k(in)g Fo(L)1246 4162 y Fx(1)1286 4198 y Fr(\()p Fj(R)1390 4162 y Fx(3)1435 4198 y Fr(\))1473 4162 y Fx(4)1546 4198 y Fr(and)h(in)f Fo(L)1918 4162 y Fx(2)1957 4198 y Fr(\()p Fj(R)2061 4162 y Fx(3)2107 4198 y Fr(\))2145 4162 y Fx(4)2218 4198 y Fr(sim)m(ultaneously)-8 b(.)45 b(Fix)32 b Fo(t)e Fn(2)g Fj(R)83 4319 y Fr(and)j(set)g Fo( )488 4334 y Fm(n)563 4319 y Fr(:=)28 b Fn(F)766 4334 y Fm(t)795 4319 y Fr(\()p Fo(\036)891 4334 y Fm(n)938 4319 y Fr(\).)43 b(Theorem)33 b(2)f(yields)g Fo( )1876 4334 y Fm(n)1951 4319 y Fn(2)d Fo(L)2112 4283 y Fx(1)2151 4319 y Fr(\()p Fj(R)2255 4283 y Fx(3)2301 4319 y Fr(\))2339 4283 y Fx(4)2400 4319 y Fn(\\)23 b(C)2541 4334 y Fy(1)2616 4319 y Fr(\()p Fj(R)2720 4283 y Fx(3)2765 4319 y Fr(\))2803 4283 y Fx(4)2843 4319 y Fr(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)926 4559 y Fn(F)1008 4508 y Fx(\(1\))998 4581 y Fm(t)1102 4559 y Fr(\()p Fo(\036)p Fr(\))82 b(=)107 b(lim)1477 4619 y Fm(n)p Fy(!1)1678 4559 y Fo( )1741 4574 y Fm(n)1816 4559 y Fn(2)28 b(C)1962 4574 y Fm(b)1997 4559 y Fr(\()p Fj(R)2101 4518 y Fx(3)2146 4559 y Fr(\))2184 4518 y Fx(4)2321 4559 y Fr(and)926 4763 y Fn(F)1008 4712 y Fx(\(2\))998 4785 y Fm(t)1102 4763 y Fr(\()p Fo(\036)p Fr(\))82 b(=)107 b(lim)1477 4822 y Fm(n)p Fy(!1)1678 4763 y Fo( )1741 4778 y Fm(n)1816 4763 y Fn(2)28 b Fo(L)1976 4721 y Fx(2)2016 4763 y Fr(\()p Fj(R)2120 4721 y Fx(3)2165 4763 y Fr(\))2203 4721 y Fx(4)2242 4763 y Fo(;)878 b Fr(\(85\))83 5026 y(where)27 b(the)e(limits)e(are)i(understo)s(o)s(d)h(in)e Fn(C)1599 5041 y Fm(b)1634 5026 y Fr(\()p Fj(R)1738 4990 y Fx(3)1784 5026 y Fr(\))1822 4990 y Fx(4)1886 5026 y Fr(and)i Fo(L)2135 4990 y Fx(2)2175 5026 y Fr(\()p Fj(R)2278 4990 y Fx(3)2324 5026 y Fr(\))2362 4990 y Fx(4)2401 5026 y Fr(,)h(resp)s(ectiv)m(ely)-8 b(.)42 b(Cho)s(ose)83 5146 y(a)32 b(subsequence)k(\()p Fo( )816 5161 y Fm(n)859 5173 y Fh(k)902 5146 y Fr(\))940 5161 y Fm(k)1015 5146 y Fr(suc)m(h)e(that)904 5388 y Fo( )967 5403 y Fm(n)1010 5415 y Fh(k)1053 5388 y Fr(\()p Fl(r)p Fr(\))27 b Fn(!)g(F)1411 5337 y Fx(\(2\))1401 5411 y Fm(t)1505 5388 y Fr(\()p Fo(\036)p Fr(\)\()p Fl(r)p Fr(\))g Fo(;)44 b(k)31 b Fn(!)c(1)h Fo(;)114 b Fr(a)p Fo(:)17 b Fr(e)p Fo(:)647 b Fr(\(86\))1653 5637 y(17)p eop %%Page: 18 18 18 17 bop 432 573 a Fr(But)32 b(since)1405 693 y Fo( )1468 708 y Fm(n)1511 720 y Fh(k)1554 693 y Fr(\()p Fl(r)p Fr(\))27 b Fn(!)g(F)1912 642 y Fx(\(1\))1902 716 y Fm(t)2006 693 y Fr(\()p Fo(\036)p Fr(\)\()p Fl(r)p Fr(\))g Fo(;)44 b(k)31 b Fn(!)c(1)p Fo(;)800 b Fr(\(87\))432 867 y(for)32 b(all)e Fl(r)p Fr(,)j(w)m(e)g(conclude)g(that)1424 1087 y Fn(F)1506 1036 y Fx(\(1\))1496 1109 y Fm(t)1600 1087 y Fr(\()p Fo(\036)p Fr(\)\()p Fl(r)p Fr(\))26 b(=)i Fn(F)2068 1036 y Fx(\(2\))2058 1109 y Fm(t)2162 1087 y Fr(\()p Fo(\036)p Fr(\)\()p Fl(r)p Fr(\))97 b(a)p Fo(:)17 b Fr(e)p Fo(:)818 b Fr(\(88\))p 3599 1306 4 66 v 3603 1244 59 4 v 3603 1306 V 3661 1306 4 66 v 432 1509 a Fl(Corollary)37 b(8.)42 b Fp(L)-5 b(et)35 b Fo(\036)27 b Fn(2)i Fo(L)1474 1473 y Fx(2)1513 1509 y Fr(\()p Fj(R)1617 1473 y Fx(3)1663 1509 y Fr(\))1701 1473 y Fx(4)1775 1509 y Fp(and)34 b Fo(K)2047 1524 y Fm(n)2122 1509 y Fr(:=)27 b Fn(f)p Fo(x)h Fn(2)g Fj(R)2545 1473 y Fx(3)2619 1509 y Fr(:)f Fn(jj)p Fo(x)p Fn(jj)2839 1538 y Fx(2)2907 1509 y Fn(\024)h Fo(n)p Fn(g)p Fp(.)44 b(Then)927 1823 y Fr(\()p Fn(F)1047 1772 y Fx(\(2\))1037 1846 y Fm(t)1141 1823 y Fo(\036)p Fr(\)\()p Fl(r)p Fr(\))27 b(=)g(l)p Fo(:)p Fr(i)p Fo(:)p Fr(m)p Fo(:)1505 1883 y Fm(n)p Fy(!1)1855 1756 y Fr(1)p 1732 1800 294 4 v 1732 1891 a(\(2)p Fo(\031)t Fr(\))1916 1863 y Fx(3)p Fm(=)p Fx(2)2107 1699 y(4)2053 1728 y Fk(X)2063 1938 y Fm(r)r Fx(=1)2213 1688 y Fk(Z)2268 1913 y Fm(K)2328 1921 y Fh(n)2391 1823 y Fo(\036)2449 1838 y Fm(r)2487 1823 y Fr(\()p Fl(p)p Fr(\))p Fo( )2688 1838 y Fi(p)p Fm(;r)2791 1823 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)3060 1782 y Fx(3)3098 1823 y Fo(p:)322 b Fr(\(89\))432 2134 y Fp(Pr)-5 b(o)g(of.)48 b Fr(Again)25 b(adapting)g(a)h(pro)s(of)f (from)g(W)-8 b(eidmann,)2445 2098 y Fx(29)2546 2134 y Fr(let)26 b Fo(\037)2742 2149 y Fm(n)2815 2134 y Fr(b)s(e)g(the)h(c)m (haracteristic)432 2254 y(function)i(of)g(the)h(set)g Fo(K)1316 2269 y Fm(n)1392 2254 y Fr(and)g(\014x)g Fo(t)e Fn(2)g Fj(R)5 b Fr(.)48 b(Then)31 b Fo(\037)2325 2269 y Fm(n)2372 2254 y Fo(\036)c Fn(2)i Fo(L)2618 2218 y Fx(1)2657 2254 y Fr(\()p Fj(R)2761 2218 y Fx(3)2807 2254 y Fr(\))2845 2218 y Fx(4)2900 2254 y Fn(\\)16 b Fo(L)3048 2218 y Fx(2)3088 2254 y Fr(\()p Fj(R)3192 2218 y Fx(3)3237 2254 y Fr(\))3275 2218 y Fx(4)3315 2254 y Fr(,)30 b(and)f(w)m(e)432 2375 y(ha)m(v)m(e)34 b Fo(\037)718 2390 y Fm(n)765 2375 y Fo(\036)27 b Fn(!)g Fo(\036)33 b Fr(in)e Fo(L)1247 2339 y Fx(2)1287 2375 y Fr(\()p Fj(R)1391 2339 y Fx(3)1437 2375 y Fr(\))1475 2339 y Fx(4)1514 2375 y Fr(.)43 b(According)33 b(to)f(Theorem)h(7,)951 2684 y(\()p Fn(F)1071 2633 y Fx(\(2\))1061 2707 y Fm(t)1165 2684 y Fr(\()p Fo(\037)1264 2699 y Fm(n)1311 2684 y Fo(\036)p Fr(\)\)\()p Fl(r)p Fr(\))27 b(=)1830 2617 y(1)p 1708 2661 V 1708 2753 a(\(2)p Fo(\031)t Fr(\))1892 2724 y Fx(3)p Fm(=)p Fx(2)2082 2560 y(4)2028 2590 y Fk(X)2038 2799 y Fm(r)r Fx(=1)2188 2549 y Fk(Z)2244 2774 y Fm(K)2304 2782 y Fh(n)2366 2684 y Fo(\036)2424 2699 y Fm(r)2462 2684 y Fr(\()p Fl(p)p Fr(\))p Fo( )2663 2699 y Fi(p)p Fm(;r)2766 2684 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))g Fo(d)3035 2643 y Fx(3)3074 2684 y Fo(p:)346 b Fr(\(90\))432 3017 y(The)33 b(claim)d(follo)m(ws)i(from)f (the)i(fact)f(that)h Fn(F)2097 2966 y Fx(\(2\))2087 3040 y Fm(t)2191 3017 y Fr(\()p Fo(\037)2290 3032 y Fm(n)2337 3017 y Fo(\036)p Fr(\))27 b Fn(!)g(F)2669 2966 y Fx(\(2\))2659 3040 y Fm(t)2763 3017 y Fr(\()p Fo(\036)p Fr(\))32 b(in)g Fo(L)3109 2981 y Fx(2)3149 3017 y Fr(\()p Fj(R)3253 2981 y Fx(3)3298 3017 y Fr(\))3336 2981 y Fx(4)3375 3017 y Fr(.)p 3599 3017 4 66 v 3603 2955 59 4 v 3603 3017 V 3661 3017 4 66 v 432 3350 a Fq(VI)t(I.)54 b(Summary)f(and)g(outlo)t(ok) 432 3569 y Fr(In)27 b(this)g(article,)f(w)m(e)i(ha)m(v)m(e)h(put)e (some)g(assertions)g(concerning)g(the)h(relativistic)c(V)-8 b(olk)m(o)m(v)432 3689 y(solutions)35 b(on)i(solid)e(mathematical)e (ground.)55 b(As)37 b(a)g(general)e(assumption,)i(w)m(e)h(to)s(ok)432 3810 y(the)25 b(electromagnetic)f(v)m(ector)i(p)s(oten)m(tial)d(to)h(b) s(e)i(con)m(tin)m(uous)f(and)g(b)s(ounded.)42 b(In)25 b(Theo-)432 3930 y(rem)g(2,)j(w)m(e)f(ha)m(v)m(e)h(justi\014ed)e(the)h (construction)f(of)g(regular)f(w)m(a)m(v)m(e)j(pac)m(k)m(ets)g(from)d (V)-8 b(olk)m(o)m(v)432 4050 y(solutions)43 b(as)h(used,)k(for)43 b(example,)k(in)1912 4014 y Fx(18)2030 4050 y Fr(and)d(ha)m(v)m(e)h (pro)m(v)m(ed)g(their)f(rapid)f(deca)m(y)i(in)432 4171 y(spatial)38 b(space.)69 b(Theorem)40 b(3)h(sho)m(ws)h(that)e(these)i (w)m(a)m(v)m(e)g(pac)m(k)m(ets)h(satisfy)d(the)h(Dirac)432 4291 y(equation,)28 b(as)g(long)e(as)i(the)g(v)m(ector)h(p)s(oten)m (tial)d(is)h(also)g Fn(C)2489 4255 y Fx(1)2556 4291 y Fr(and)h(has)g(a)f(b)s(ounded)i(deriv)-5 b(a-)432 4412 y(tiv)m(e.)91 b(The)50 b(\\normalization")44 b(prop)s(ert)m(y)49 b(is)f(v)m(eri\014ed)i(in)e(the)h(cen)m(tral)f(Theorem)h(5)432 4532 y(and)34 b(allo)m(ws)e(the)j(de\014nition)d(of)i(w)m(a)m(v)m(e)h (pac)m(k)m(ets)h(for)e(an)m(y)g Fo(L)2618 4496 y Fx(2)2692 4532 y Fr(function)f(in)g(momen)m(tum)432 4652 y(space.)62 b(Finally)-8 b(,)37 b(Theorem)i(7)f(pro)m(vides)h(a)g(useful)f(in)m (tegral)f(represen)m(tation)i(for)f(the)432 4773 y(momen)m(tum-space)32 b(functions)g(con)m(tained)h(in)f Fo(L)2247 4737 y Fx(1)2309 4773 y Fn(\\)22 b Fo(L)2463 4737 y Fx(2)2503 4773 y Fr(.)578 4893 y(As)39 b(men)m(tioned)g(in)f(the)h(in)m(tro)s(duction,)g(it)f(is) g(still)e(desirable)i(to)h(\014nd)g(a)g(rigorous)432 5013 y(pro)s(of)33 b(of)g(the)i(\\completeness")f(of)g(the)g(V)-8 b(olk)m(o)m(v)35 b(solutions,)e(i.)g(e.)48 b(for)34 b(the)g(h)m(yp)s (othesis)432 5134 y(that)24 b(for)f(an)m(y)i(initial)20 b(w)m(a)m(v)m(e)26 b(pac)m(k)m(et)g Fo( )1815 5149 y Fx(0)1882 5134 y Fn(2)i Fo(L)2042 5098 y Fx(2)2082 5134 y Fr(\()p Fj(R)2186 5098 y Fx(3)2231 5134 y Fr(\))2269 5098 y Fx(4)2333 5134 y Fr(at)23 b(time)g Fo(t)28 b Fr(=)g(0,)d(there)g (is)e(a)h(momen-)432 5254 y(tum)39 b(distribution)g Fo(\036)h Fn(2)i Fo(L)1457 5218 y Fx(2)1497 5254 y Fr(\()p Fj(R)1601 5218 y Fx(3)1646 5254 y Fr(\))1684 5218 y Fx(4)1764 5254 y Fr(from)d(whic)m(h)i Fo( )2352 5269 y Fx(0)2432 5254 y Fr(and)f(also)g(the)g(time-dep)s(enden)m(t)432 5388 y(solution)33 b Fo( )t Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))34 b(with)g Fo( )t Fr(\()p Fn(\001)p Fo(;)17 b Fr(0\))30 b(=)h Fo( )1793 5403 y Fx(0)1867 5388 y Fr(follo)m(ws.)48 b(In)35 b(other)g(w)m(ords,)h(the)f(mapping)e Fn(F)3576 5337 y Fx(\(2\))3566 5411 y Fm(t)2002 5637 y Fr(18)p eop %%Page: 19 19 19 18 bop 83 573 a Fr(from)24 b(\(82\))g(should)g(b)s(e)h(surjectiv)m (e)h(and)f(therefore)h(unitary)e(for)g(a)h(\\reasonable")f(v)m(ector)83 693 y(p)s(oten)m(tial)511 669 y Fl(~)497 693 y(A)o Fr(.)49 b(A)34 b(comparison)f(with)h(the)h(\014eld-free)f(case)2298 669 y Fl(~)2283 693 y(A)c Fn(\021)i Fr(0)i(suggests)h(that)f(the)83 814 y(in)m(v)m(erse)1384 981 y Fn(G)1443 996 y Fm(t)1500 981 y Fr(:=)1631 870 y Fk(\020)1690 981 y Fn(F)1772 930 y Fx(\(2\))1762 1003 y Fm(t)1866 870 y Fk(\021)1926 893 y Fy(\000)p Fx(1)3147 981 y Fr(\(91\))83 1210 y(migh)m(t)d(b)s(e)i (obtainable)e(b)m(y)i(the)g(form)m(ula)161 1542 y Fo(\036)p Fr(\()p Fl(p)p Fr(\))27 b(:=)h(\()p Fn(G)612 1557 y Fm(t)642 1542 y Fr(\()p Fo( )t Fr(\)\))16 b(\()p Fl(p)p Fr(\))28 b(=)1109 1372 y Fk( )1320 1475 y Fr(1)p 1197 1519 294 4 v 1197 1611 a(\(2)p Fo(\031)t Fr(\))1381 1582 y Fx(3)p Fm(=)p Fx(2)1572 1418 y(4)1518 1448 y Fk(X)1528 1657 y Fm(r)r Fx(=1)1678 1407 y Fk(Z)1733 1632 y Fe(R)1781 1613 y Fg(3)1837 1542 y Fn(h)p Fo( )1939 1557 y Fi(p)p Fm(;r)2042 1542 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fo(;)g( )t Fr(\()p Fl(r)p Fr(\))p Fn(i)32 b Fo(d)2598 1501 y Fx(3)2637 1542 y Fo(p)2686 1372 y Fk(!)2765 1671 y Fm(r)r Fx(=1)p Fm(;:::)n(;)p Fx(4)3043 1542 y Fo(;)77 b Fr(\(92\))83 1889 y(at)40 b(least)g(when)h Fo( )k Fn(2)c(C)981 1853 y Fy(1)975 1913 y Fx(0)1056 1889 y Fr(.)66 b(The)42 b(\\ph)m(ysical")d(v)m(ersion)i(of)f(the)g(ab)s(o)m(v)m(e)h(in)m(v)m (ersion)g(for-)83 2009 y(m)m(ula)31 b(\(92\))h(is)723 2223 y Fx(4)669 2253 y Fk(X)679 2462 y Fm(r)r Fx(=1)829 2212 y Fk(Z)884 2437 y Fe(R)932 2418 y Fg(3)988 2347 y Fo( )1055 2306 y Fy(y)1051 2372 y Fi(p)p Fm(;r)1154 2347 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fo( )1418 2362 y Fi(p)p Fm(;r)1521 2347 y Fr(\()p Fl(r)1605 2306 y Fy(0)1628 2347 y Fo(;)g(t)p Fr(\))g Fo(d)1813 2306 y Fx(3)1851 2347 y Fo(p)28 b Fr(=)f(\(2)p Fo(\031)t Fr(\))2215 2306 y Fx(3)2254 2347 y Fo(\016)2301 2306 y Fx(\(3\))2396 2347 y Fr(\()p Fl(r)22 b Fn(\000)g Fl(r)2647 2306 y Fy(0)2670 2347 y Fr(\))p Fo(:)412 b Fr(\(93\))83 2682 y(The)24 b(v)-5 b(alidit)m(y)20 b(of)j(the)g(in)m(v)m(ersion)g(form)m(ula)d (\(92\))j(w)m(ould)f(allo)m(w)f(one)i(to)g(write)f(the)h(unitary)83 2803 y(propagator)32 b(for)g(the)h(Dirac)e(equation)h(\(30\))g(as)1287 3064 y Fo(U)10 b Fr(\()p Fo(t;)17 b(t)1515 3023 y Fy(0)1539 3064 y Fr(\))28 b(=)f Fn(F)1790 3013 y Fx(\(2\))1780 3087 y Fm(t)1906 3064 y Fn(\016)22 b(G)2037 3079 y Fm(t)2062 3060 y Fc(0)2090 3064 y Fo(:)1030 b Fr(\(94\))83 3312 y(This)32 b(conjecture)h(is)f(used)h(b)m(y)f(San)g(Rom\023)-49 b(an)31 b(and)h(co)m(w)m(ork)m(ers)i(in)2468 3276 y Fx(18{21)2680 3312 y Fr(for)d(the)i(calcula-)83 3432 y(tion)d(of)h(the)h(time)e(ev)m (olution)g(of)h(a)g(V)-8 b(olk)m(o)m(v)31 b(w)m(a)m(v)m(e)i(pac)m(k)m (et.)45 b(Ha)m(ving)31 b(established)g(\(92\),)83 3553 y(one)41 b(migh)m(t)d(also)i(try)g(to)g(understand)i(on)e(a)g(more)g (analytical)e(lev)m(el)i(the)h(squeezing)83 3673 y(and)33 b(distortion)e(of)h(w)m(a)m(v)m(e)i(pac)m(k)m(ets)h(observ)m(ed)f(in.) 1922 3637 y Fx(18,)9 b(19)229 3803 y Fr(In)35 b(order)f(to)g(pro)m(v)m (e)h(\(92\),)f(one)h(should)f(\(i\))f(v)m(erify)h(the)h(existence)g(of) f(the)g(in)m(tegral)83 3923 y(for)f Fo( )h Fn(2)c(C)484 3887 y Fy(1)478 3948 y Fx(0)559 3923 y Fr(,)k(\(ii\))e(sho)m(w)j(that)e Fo(\036)d Fn(2)g Fo(L)1489 3887 y Fx(1)1552 3923 y Fn(\\)23 b Fo(L)1707 3887 y Fx(2)1747 3923 y Fr(,)34 b(and)g(\(iii\))d (demonstrate)i(with)h(the)g(help)83 4044 y(of)e(Theorem)h(7)f(that)1331 4192 y Fo( )g Fr(=)27 b Fn(F)1611 4141 y Fx(\(2\))1601 4214 y Fm(t)1722 4192 y Fr(\()p Fn(G)1819 4207 y Fm(t)1848 4192 y Fr(\()p Fo( )t Fr(\)\))17 b Fo(:)1074 b Fr(\(95\))83 4399 y(There)30 b(are)g(strong)f(indications)e(to)i(the)h(author)f (that)g(the)g(assertions)h(\(i\))e(and)i(\(ii\))d(are)83 4519 y(correct)i(for)e(a)h(b)s(ounded)g(v)m(ector)h(p)s(oten)m(tial) 1728 4495 y Fl(~)1714 4519 y(A)e Fn(2)h(C)1978 4483 y Fx(2)2046 4519 y Fr(with)f(b)s(ounded)i(\014rst)f(and)h(second)83 4640 y(deriv)-5 b(ates.)43 b(More)33 b(details)e(and)h(results)g(shall) f(b)s(e)h(supplied)g(in)f(a)h(future)g(publication.)83 5030 y Fq(Ac)l(kno)l(wledgmen)l(t)83 5268 y Fr(The)g(author)f(is)g (grateful)f(to)h(Carsten)i(M)s(\177)-51 b(uller)29 b(for)i(v)-5 b(aluable)30 b(discussions)i(and)f(for)g(his)83 5388 y(p)s(oin)m(ting)g(out)h(sev)m(eral)i(mistak)m(es)e(in)g(a)g(previous)h (v)m(ersion)g(of)g(this)f(man)m(uscript.)1653 5637 y(19)p eop %%Page: 20 20 20 19 bop 432 573 a Fq(App)t(endix)53 b(A:)h(Pro)t(of)g(of)g(Theorem)e (5)432 829 y Fr(In)33 b(this)f(app)s(endix,)h(w)m(e)g(pro)m(vide)g(the) g(pro)s(of)f(for)g(the)h(main)e(Theorem)i(5.)432 1205 y Fp(Pr)-5 b(o)g(of.)48 b Fr(Fix)32 b Fo(t)c Fn(2)g Fj(R)43 b Fr(and)33 b Fo(\036)27 b Fn(2)i(C)1601 1169 y Fy(1)1595 1229 y Fx(0)1676 1205 y Fr(\()p Fj(R)1779 1169 y Fx(3)1825 1205 y Fr(\))1863 1169 y Fx(4)1902 1205 y Fr(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(to)e(sho)m(w)h(that)868 1569 y Fn(jj)o(F)995 1584 y Fm(t)1025 1569 y Fr(\()p Fo(\036)p Fr(\))p Fn(j)o(j)1214 1521 y Fx(2)1214 1598 y Fm(L)1262 1579 y Fg(2)1296 1598 y Fx(\()p Fe(R)1371 1579 y Fg(3)1406 1598 y Fx(\))1433 1579 y Fg(4)1500 1569 y Fn(\021)1702 1502 y Fr(1)p 1615 1546 223 4 v 1615 1638 a(\(2)p Fo(\031)t Fr(\))1799 1609 y Fx(3)1945 1445 y(4)1891 1475 y Fk(X)1865 1687 y Fm(r)n(;r)1949 1668 y Fc(0)1970 1687 y Fx(=1)2077 1434 y Fk(Z)2133 1659 y Fe(R)2181 1640 y Fg(3)2236 1569 y Fo(I)2279 1584 y Fm(r)n(;r)2363 1565 y Fc(0)2389 1569 y Fr(\()p Fl(r)p Fr(\))27 b Fo(d)2589 1528 y Fx(3)2628 1569 y Fo(r)j Fr(=)e Fn(jj)o Fo(\036)p Fn(jj)2974 1521 y Fx(2)2974 1598 y Fm(L)3022 1579 y Fg(2)3057 1598 y Fx(\()p Fe(R)3132 1579 y Fg(3)3167 1598 y Fx(\))3194 1579 y Fg(4)3472 1569 y Fr(\(A1\))432 1942 y(with)639 2239 y Fo(I)682 2254 y Fm(r)n(;r)766 2235 y Fc(0)792 2239 y Fr(\()p Fl(r)p Fr(\))f(:=)1072 2104 y Fk(Z)1127 2329 y Fe(R)1175 2310 y Fg(3)1231 2104 y Fk(Z)1286 2329 y Fe(R)1334 2310 y Fg(3)p 1389 2153 234 4 v 1389 2239 a Fo(\036)1447 2254 y Fm(r)1485 2239 y Fr(\()p Fl(p)p Fr(\))p Fo(\036)1681 2254 y Fm(r)1715 2235 y Fc(0)1741 2239 y Fr(\()p Fl(p)1841 2198 y Fy(0)1865 2239 y Fr(\))17 b Fn(h)o Fo( )2021 2254 y Fi(p)p Fm(;r)2124 2239 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))p Fo(;)g( )2432 2254 y Fi(p)2477 2235 y Fc(0)2500 2254 y Fm(r)2534 2235 y Fc(0)2560 2239 y Fr(\()p Fl(r)p Fo(;)g(t)p Fr(\))p Fn(i)2799 2271 y Fe(C)2841 2253 y Fg(4)2919 2239 y Fo(d)2970 2198 y Fx(3)3009 2239 y Fo(p)g(d)3126 2198 y Fx(3)3165 2239 y Fo(p)3214 2198 y Fy(0)3237 2239 y Fo(;)208 b Fr(\(A2\))432 2561 y(where)25 b(the)f(bar)f(denotes)i(complex)f(conjugation.)39 b(Note)24 b(that)f(all)f(in)m(tegrals)h(app)s(earing)432 2681 y(ab)s(o)m(v)m(e)33 b(exist)g(since)g Fn(F)1250 2696 y Fm(t)1279 2681 y Fr(\()p Fo(\036)p Fr(\))27 b Fn(2)i Fo(L)1601 2645 y Fx(2)1640 2681 y Fr(\()p Fj(R)1744 2645 y Fx(3)1790 2681 y Fr(\))1828 2645 y Fx(4)1900 2681 y Fr(b)m(y)k(virtue)g(of)f(Theorem)h(2.)578 2820 y(Inserting)g(the)g(V) -8 b(olk)m(o)m(v)32 b(solutions)g(\(9\),)g(w)m(e)i(\014nd)587 3142 y Fo(I)630 3157 y Fm(r)n(;r)714 3138 y Fc(0)740 3142 y Fr(\()p Fl(r)p Fr(\))83 b(=)1104 3007 y Fk(Z)1159 3232 y Fe(R)1207 3213 y Fg(3)1262 3007 y Fk(Z)1318 3232 y Fe(R)1366 3213 y Fg(3)p 1421 3056 V 1421 3142 a Fo(\036)1479 3157 y Fm(r)1517 3142 y Fr(\()p Fl(p)p Fr(\))p Fo(\036)1713 3157 y Fm(r)1747 3138 y Fc(0)1773 3142 y Fr(\()p Fl(p)1873 3101 y Fy(0)1896 3142 y Fr(\))17 b(exp)2117 3062 y Fk(\002)2158 3142 y Fo(i)2208 3062 y Fk(\000)2254 3142 y Fo(S)2320 3101 y Fx(\()p Fm(\017)2376 3120 y Fh(r)2407 3106 y Fc(0)2432 3101 y Fx(\))2464 3142 y Fr(\()p Fl(p)2564 3101 y Fy(0)2588 3142 y Fo(;)g Fl(r)p Fo(;)g(t)p Fr(\))k Fn(\000)i Fo(S)2982 3101 y Fx(\()p Fm(\017)3038 3109 y Fh(r)3072 3101 y Fx(\))3104 3142 y Fr(\()p Fl(p)p Fo(;)17 b Fl(r)p Fo(;)g(t)p Fr(\))3411 3062 y Fk(\001)o(\003)1104 3357 y Fn(\002)1198 3276 y Fk(\012)1245 3357 y Fl(M)1351 3315 y Fx(\()p Fm(\017)1407 3323 y Fh(r)1442 3315 y Fx(\))1473 3357 y Fr(\()p Fl(p)p Fo(;)g(t)23 b Fn(\000)f Fo(z)t Fr(\))p Fo(u)1917 3372 y Fm(r)1955 3357 y Fr(\()p Fl(p)p Fr(\))p Fo(;)17 b Fl(M)2243 3315 y Fx(\()p Fm(\017)2299 3334 y Fh(r)2330 3320 y Fc(0)2357 3315 y Fx(\))2388 3357 y Fr(\()p Fl(p)2488 3315 y Fy(0)2512 3357 y Fo(;)g(t)22 b Fn(\000)g Fo(z)t Fr(\))p Fo(u)2855 3372 y Fm(r)2889 3353 y Fc(0)2916 3357 y Fr(\()p Fl(p)3016 3315 y Fy(0)3039 3357 y Fr(\))3077 3276 y Fk(\013)3158 3357 y Fo(d)3209 3315 y Fx(3)3248 3357 y Fo(p)17 b(d)3365 3315 y Fx(3)3403 3357 y Fo(p)3452 3315 y Fy(0)945 3560 y Fr(=)1104 3425 y Fk(Z)1159 3650 y Fe(R)1207 3631 y Fg(3)1262 3425 y Fk(Z)1318 3650 y Fe(R)1366 3631 y Fg(3)p 1421 3474 V 1421 3560 a Fo(\036)1479 3575 y Fm(r)1517 3560 y Fr(\()p Fl(p)p Fr(\))p Fo(\036)1713 3575 y Fm(r)1747 3556 y Fc(0)1773 3560 y Fr(\()p Fl(p)1873 3519 y Fy(0)1896 3560 y Fr(\))g(exp)h([)p Fo(i)f Fr(\()p Fo(\017)2271 3575 y Fm(r)2309 3560 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(\017)2664 3575 y Fm(r)2698 3556 y Fc(0)2725 3560 y Fo(\024)p Fr(\()p Fl(p)2881 3519 y Fy(0)2904 3560 y Fr(\)\))17 b Fo(t)p Fr(])1104 3774 y Fn(\002)1198 3694 y Fk(\012)1245 3774 y Fl(M)1351 3733 y Fx(\()p Fm(\017)1407 3741 y Fh(r)1442 3733 y Fx(\))1473 3774 y Fr(\()p Fl(p)p Fo(;)g(t)23 b Fn(\000)f Fo(z)t Fr(\))p Fo(u)1917 3789 y Fm(r)1955 3774 y Fr(\()p Fl(p)p Fr(\))p Fo(;)17 b Fl(M)2243 3733 y Fx(\()p Fm(\017)2299 3752 y Fh(r)2330 3738 y Fc(0)2357 3733 y Fx(\))2388 3774 y Fr(\()p Fl(p)2488 3733 y Fy(0)2512 3774 y Fo(;)g(t)22 b Fn(\000)g Fo(z)t Fr(\))p Fo(u)2855 3789 y Fm(r)2889 3770 y Fc(0)2916 3774 y Fr(\()p Fl(p)3016 3733 y Fy(0)3039 3774 y Fr(\))3077 3694 y Fk(\013)1104 3957 y Fn(\002)17 b Fr(exp)1363 3846 y Fk(h)1410 3957 y Fo(i)1460 3846 y Fk(\020)1520 3957 y Fl(q)1579 3904 y Fx(\()p Fm(\017)1635 3923 y Fh(r)1666 3909 y Fc(0)1692 3904 y Fx(\))1579 3979 y Fm(t)p Fy(\000)p Fm(z)1724 3957 y Fr(\()p Fl(p)1824 3916 y Fy(0)1847 3957 y Fr(\))22 b Fn(\000)h Fl(q)2066 3906 y Fx(\()p Fm(\017)2122 3914 y Fh(r)2157 3906 y Fx(\))2066 3979 y Fm(t)p Fy(\000)p Fm(z)2188 3957 y Fr(\()p Fl(p)p Fr(\))2326 3846 y Fk(\021)2408 3957 y Fn(\001)f Fl(s)2502 3972 y Fm(t)2532 3957 y Fr(\()p Fl(r)p Fr(\))2654 3846 y Fk(i)2734 3957 y Fo(d)2785 3916 y Fx(3)2824 3957 y Fo(p)17 b(d)2941 3916 y Fx(3)2979 3957 y Fo(p)3028 3916 y Fy(0)3052 3957 y Fo(:)393 b Fr(\(A3\))432 4259 y(Note)33 b(that)g(the)h(only)f Fo(x)h Fr(and)g Fo(y)i Fr(dep)s(endence)g(is)d(con)m(tained)g(in)g(the)h(term)f Fl(s)3255 4274 y Fm(t)3285 4259 y Fr(\()p Fl(r)p Fr(\).)45 b(This)432 4379 y(fact)26 b(is)g(in)m(timately)d(connected)28 b(with)e(the)h(v)-5 b(alidit)m(y)24 b(of)i(the)h(eigen)m(v)-5 b(alue)26 b(equations)g(\(27\))432 4499 y(and)41 b(\(28\).)70 b(W)-8 b(e)42 b(th)m(us)h(p)s(erform)d(the)i(in)m(tegration)e(o)m(v)m (er)j(these)g(t)m(w)m(o)f(v)-5 b(ariables)40 b Fl(r)3490 4514 y Fy(?)3592 4499 y Fn(\021)432 4620 y Fr(\()p Fo(x;)17 b(y)t Fr(\))31 b(\014rst)i(and)g(obtain)785 4846 y Fk(Z)840 5071 y Fe(R)888 5052 y Fg(2)944 4981 y Fo(I)987 4996 y Fm(r)n(;r)1071 4977 y Fc(0)1097 4981 y Fr(\()p Fl(r)p Fr(\))27 b Fo(d)1297 4940 y Fx(2)1336 4981 y Fo(r)1380 4996 y Fy(?)1522 4981 y Fr(=)1681 4846 y Fk(Z)1736 5071 y Fe(R)1784 5052 y Fg(2)1840 4811 y Fk( )1935 4846 y(Z)1990 5071 y Fe(R)2038 5052 y Fg(2)2094 4846 y Fk(Z)2149 5071 y Fe(R)2197 5052 y Fg(2)2252 4871 y Fk(D)2326 4955 y Fr(~)2313 4981 y Fo(\036)2371 4996 y Fm(r)2409 4981 y Fr(\()p Fl(p)2509 4996 y Fy(?)2568 4981 y Fo(;)17 b(z)t Fr(\))p Fo(;)2756 4955 y Fr(~)2743 4981 y Fo(\036)2801 4996 y Fm(r)2835 4977 y Fc(0)2861 4981 y Fr(\()p Fl(p)2961 4940 y Fy(0)2961 5006 y(?)3021 4981 y Fo(;)g(z)t Fr(\))3152 4871 y Fk(E)3213 5050 y Fe(C)3255 5031 y Fg(4)1681 5313 y Fn(\002)p Fo(e)1803 5272 y Fm(i)p Fx(\()p Fm(\017)1883 5291 y Fh(r)1914 5277 y Fc(0)1941 5272 y Fi(p)1986 5249 y Fc(0)1986 5295 y(?)2038 5272 y Fy(\000)p Fm(\017)2122 5280 y Fh(r)2156 5272 y Fi(p)2201 5284 y Fc(?)2253 5272 y Fx(\))p Fy(\001)p Fi(r)2334 5284 y Fc(?)2407 5313 y Fo(d)2458 5272 y Fx(2)2497 5313 y Fo(p)2546 5328 y Fy(?)2622 5313 y Fo(d)2673 5272 y Fx(2)2712 5313 y Fo(p)2761 5272 y Fy(0)2761 5338 y(?)2820 5143 y Fk(!)2926 5313 y Fo(d)2977 5272 y Fx(2)3016 5313 y Fo(r)3060 5328 y Fy(?)3119 5313 y Fo(;)326 b Fr(\(A4\))2002 5637 y(20)p eop %%Page: 21 21 21 20 bop 83 573 a Fr(where)187 769 y(~)174 795 y Fo(\036)232 810 y Fm(r)270 795 y Fr(\()p Fl(p)370 810 y Fy(?)429 795 y Fo(;)17 b(z)t Fr(\))83 b(:=)829 660 y Fk(Z)885 885 y Fe(R)953 795 y Fo(\036)1011 810 y Fm(r)1049 795 y Fr(\()p Fl(p)p Fr(\))p Fo( )1250 810 y Fi(p)p Fm(;r)1353 795 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fo(e)1599 754 y Fy(\000)p Fm(i\017)1707 762 y Fh(r)1741 754 y Fi(p)1786 766 y Fc(?)1839 754 y Fy(\001)p Fi(r)1892 771 y Fc(?)1965 795 y Fo(dp)2065 810 y Fm(z)657 1054 y Fr(=)829 918 y Fk(Z)885 1144 y Fe(R)953 1054 y Fo(\036)1011 1069 y Fm(r)1049 1054 y Fr(\()p Fl(p)p Fr(\))p Fl(M)1293 1013 y Fx(\()p Fm(\017)1349 1021 y Fh(r)1384 1013 y Fx(\))1416 1054 y Fr(\()p Fl(p)p Fo(;)g(t)22 b Fn(\000)h Fo(z)t Fr(\))p Fo(u)1860 1069 y Fm(r)1898 1054 y Fr(\()p Fl(p)p Fr(\))p Fo(e)2081 1013 y Fy(\000)p Fm(i\017)2189 1021 y Fh(r)2223 1013 y Fm(\024)p Fx(\()p Fi(p)p Fx(\))p Fm(t)829 1323 y Fn(\002)17 b Fr(exp)1089 1183 y Fk(\024)1142 1323 y Fo(i)1185 1256 y(\017)1224 1271 y Fm(r)p 1185 1301 78 4 v 1199 1392 a Fr(2)1289 1183 y Fk(\022)1372 1256 y Fo(m)1457 1220 y Fx(2)1519 1256 y Fr(+)22 b Fo(\031)1672 1271 y Fy(?)1731 1256 y Fr(\()p Fo(t)h Fn(\000)f Fo(z)t(;)17 b(\017)2058 1271 y Fm(r)2097 1256 y Fl(p)2159 1271 y Fy(?)2218 1256 y Fr(\))2256 1220 y Fx(2)p 1372 1301 924 4 v 1737 1392 a Fo(\024)p Fr(\()p Fl(p)p Fr(\))2328 1323 y Fn(\000)22 b Fo(\024)p Fr(\()p Fl(p)p Fr(\))2621 1183 y Fk(\023)2712 1323 y Fr(\()p Fo(z)k Fn(\000)d Fo(t)p Fr(\))2994 1183 y Fk(\025)3063 1323 y Fo(dp)3163 1338 y Fm(z)3203 1323 y Fo(:)3123 1526 y Fr(\(A5\))83 1725 y(Since)33 b Fo(\036)396 1740 y Fm(r)461 1725 y Fn(2)28 b(C)613 1689 y Fy(1)607 1749 y Fx(0)688 1725 y Fr(\()p Fj(R)792 1689 y Fx(3)838 1725 y Fr(\),)k(w)m(e)i(see)f(that)g(for)f(ev) m(ery)i Fo(z)t Fr(,)1121 1897 y(~)1108 1924 y Fo(\036)1166 1939 y Fm(r)1204 1924 y Fr(\()p Fn(\001)p Fo(;)17 b(z)t Fr(\))27 b Fn(2)i(C)1581 1883 y Fy(1)1575 1948 y Fx(0)1656 1924 y Fr(\()p Fj(R)1759 1883 y Fx(2)1805 1924 y Fr(\))e Fn(\032)i Fo(L)2042 1883 y Fx(2)2081 1924 y Fr(\()p Fj(R)2185 1883 y Fx(2)2231 1924 y Fr(\))p Fo(:)827 b Fr(\(A6\))83 2122 y(Using)32 b(P)m(arsev)-5 b(al's)33 b(iden)m(tit)m(y)-8 b(,)33 b(w)m(e)g(\014nd)264 2229 y Fk(Z)320 2454 y Fe(R)368 2435 y Fg(2)423 2364 y Fo(I)466 2379 y Fm(r)n(;r)550 2360 y Fc(0)576 2364 y Fr(\()p Fl(r)p Fr(\))27 b Fo(d)776 2323 y Fx(2)815 2364 y Fo(r)859 2379 y Fy(?)946 2364 y Fr(=)h(\(2)p Fo(\031)t Fr(\))1234 2323 y Fx(2)1289 2229 y Fk(Z)1345 2454 y Fe(R)1393 2435 y Fg(2)1448 2254 y Fk(D)1521 2338 y Fr(~)1509 2364 y Fo(\036)1567 2379 y Fm(r)1604 2364 y Fr(\()p Fo(\017)1681 2379 y Fm(r)1720 2364 y Fl(p)1782 2379 y Fy(?)1841 2364 y Fo(;)17 b(z)t Fr(\))p Fo(;)2029 2338 y Fr(~)2016 2364 y Fo(\036)2074 2379 y Fm(r)2108 2360 y Fc(0)2134 2364 y Fr(\()p Fo(\017)2211 2379 y Fm(r)2245 2360 y Fc(0)2272 2364 y Fl(p)2334 2379 y Fy(?)2393 2364 y Fo(;)g(z)t Fr(\))2524 2254 y Fk(E)2585 2433 y Fe(C)2627 2414 y Fg(4)2716 2364 y Fo(d)2767 2323 y Fx(2)2807 2364 y Fo(p)2856 2379 y Fy(?)2915 2364 y Fo(:)181 b Fr(\(A7\))83 2610 y(Moreo)m(v)m(er,)36 b(with)d(the)h(metho) s(d)f(applied)g(in)g(the)h(pro)s(of)f(of)g(Theorem)h(2,)g(w)m(e)h(can)f (ev)m(en)83 2730 y(sho)m(w)g(that)1446 2825 y(~)1433 2851 y Fo(\036)1491 2866 y Fm(r)1557 2851 y Fn(2)28 b Fo(L)1717 2810 y Fx(2)1756 2851 y Fr(\()p Fj(R)1860 2810 y Fx(3)1906 2851 y Fr(\))p Fo(:)1152 b Fr(\(A8\))83 3016 y(W)-8 b(e)31 b(ma)m(y)g(therefore)g(in)m(terc)m(hange)g(the)g(order)g (of)f(in)m(tegrations)g(b)m(y)h(F)-8 b(ubini's)30 b(theorem:)383 3215 y Fn(j)o(jF)510 3230 y Fm(t)539 3215 y Fr(\()p Fo(\036)p Fr(\))p Fn(j)o(j)728 3244 y Fm(L)776 3225 y Fg(2)811 3244 y Fx(\()p Fe(R)886 3225 y Fg(3)921 3244 y Fx(\))948 3225 y Fg(4)466 3471 y Fr(=)664 3404 y(1)p 634 3448 108 4 v 634 3539 a(2)p Fo(\031)849 3347 y Fx(4)795 3376 y Fk(X)769 3589 y Fm(r)n(;r)853 3570 y Fc(0)874 3589 y Fx(=1)981 3335 y Fk(Z)1036 3561 y Fe(R)1105 3331 y Fk(\022)1179 3335 y(Z)1234 3561 y Fe(R)1282 3542 y Fg(2)1337 3360 y Fk(D)1411 3445 y Fr(~)1398 3471 y Fo(\036)1456 3486 y Fm(r)1494 3471 y Fr(\()p Fo(\017)1571 3486 y Fm(r)1609 3471 y Fl(p)1671 3486 y Fy(?)1730 3471 y Fo(;)17 b(z)t Fr(\))p Fo(;)1918 3445 y Fr(~)1905 3471 y Fo(\036)1963 3486 y Fm(r)1997 3467 y Fc(0)2024 3471 y Fr(\()p Fo(\017)2101 3486 y Fm(r)2135 3467 y Fc(0)2161 3471 y Fl(p)2223 3486 y Fy(?)2282 3471 y Fo(;)g(z)t Fr(\))2413 3360 y Fk(E)2475 3540 y Fe(C)2517 3521 y Fg(4)2606 3471 y Fo(d)2657 3430 y Fx(2)2696 3471 y Fo(p)2745 3486 y Fy(?)2804 3331 y Fk(\023)2894 3471 y Fo(dz)466 3822 y Fr(=)664 3755 y(1)p 634 3800 V 634 3891 a(2)p Fo(\031)849 3698 y Fx(4)795 3728 y Fk(X)769 3940 y Fm(r)n(;r)853 3921 y Fc(0)874 3940 y Fx(=1)981 3687 y Fk(Z)1036 3912 y Fe(R)1084 3893 y Fg(2)1140 3682 y Fk(\022)1213 3687 y(Z)1268 3912 y Fe(R)1337 3712 y Fk(D)1411 3796 y Fr(~)1398 3822 y Fo(\036)1456 3837 y Fm(r)1494 3822 y Fr(\()p Fo(\017)1571 3837 y Fm(r)1609 3822 y Fl(p)1671 3837 y Fy(?)1730 3822 y Fo(;)g(z)t Fr(\))p Fo(;)1918 3796 y Fr(~)1905 3822 y Fo(\036)1963 3837 y Fm(r)1997 3818 y Fc(0)2024 3822 y Fr(\()p Fo(\017)2101 3837 y Fm(r)2135 3818 y Fc(0)2161 3822 y Fl(p)2223 3837 y Fy(?)2282 3822 y Fo(;)g(z)t Fr(\))2413 3712 y Fk(E)2475 3891 y Fe(C)2517 3872 y Fg(4)2606 3822 y Fo(dz)2706 3682 y Fk(\023)2796 3822 y Fo(d)2847 3781 y Fx(2)2886 3822 y Fo(p)2935 3837 y Fy(?)2994 3822 y Fo(:)102 b Fr(\(A9\))83 4125 y(The)32 b(iden)m(tit)m(y)h(\(A1\){i.)d(e.)43 b(the)32 b(statemen)m(t)g(of)f(the)g(theorem{is)g(true)g(if)g(for)f(ev)m(ery)j Fl(p)3234 4140 y Fy(?)3294 4125 y Fr(,)205 4329 y(1)p 176 4374 V 176 4465 a(2)p Fo(\031)310 4261 y Fk(Z)366 4487 y Fe(R)434 4286 y Fk(D)508 4371 y Fr(~)495 4397 y Fo(\036)553 4412 y Fm(r)591 4397 y Fr(\()p Fo(\017)668 4412 y Fm(r)706 4397 y Fl(p)768 4412 y Fy(?)827 4397 y Fo(;)17 b(z)t Fr(\))p Fo(;)1015 4371 y Fr(~)1002 4397 y Fo(\036)1060 4412 y Fm(r)1094 4393 y Fc(0)1121 4397 y Fr(\()p Fo(\017)1198 4412 y Fm(r)1232 4393 y Fc(0)1258 4397 y Fl(p)1320 4412 y Fy(?)1380 4397 y Fo(;)g(z)t Fr(\))1511 4286 y Fk(E)1572 4466 y Fe(C)1614 4447 y Fg(4)1703 4397 y Fo(dz)32 b Fn(\021)484 4588 y Fr(1)p 455 4633 V 455 4724 a(2)p Fo(\031)589 4545 y Fk(D)663 4629 y Fr(~)650 4656 y Fo(\036)708 4671 y Fm(r)746 4656 y Fr(\()p Fo(\017)823 4671 y Fm(r)861 4656 y Fl(p)923 4671 y Fy(?)982 4656 y Fo(;)17 b Fn(\001)p Fr(\))p Fo(;)1148 4629 y Fr(~)1136 4656 y Fo(\036)1194 4671 y Fm(r)1228 4652 y Fc(0)1253 4656 y Fr(\()p Fo(\017)1330 4671 y Fm(r)1364 4652 y Fc(0)1391 4656 y Fl(p)1453 4671 y Fy(?)1512 4656 y Fo(;)g Fn(\001)p Fr(\))1622 4545 y Fk(E)1682 4725 y Fm(L)1730 4706 y Fg(2)1797 4656 y Fr(=)27 b Fo(\016)1943 4671 y Fm(r)n(;r)2027 4652 y Fc(0)2070 4520 y Fk(Z)2125 4746 y Fe(R)2194 4656 y Fn(j)p Fo(\036)2280 4671 y Fm(r)2317 4656 y Fr(\()p Fo(\017)2394 4671 y Fm(r)2433 4656 y Fl(p)2495 4671 y Fy(?)2554 4656 y Fo(;)17 b(p)2647 4671 y Fm(z)2687 4656 y Fr(\))p Fn(j)2752 4607 y Fx(2)2825 4656 y Fo(dp)2925 4671 y Fm(z)2964 4656 y Fo(:)83 b Fr(\(A10\))229 4907 y(F)-8 b(or)28 b(the)g(rest)h(of)e(the) i(pro)s(of,)f(w)m(e)h(shall)e(\014x)i Fl(p)1856 4922 y Fy(?)1943 4907 y Fr(and)f(v)m(erify)h(\(A10\).)41 b(In)29 b(preparation)83 5027 y(for)j(exploiting)f(the)i(symmetry)f(\(32\))g (of)g(the)h(V)-8 b(olk)m(o)m(v)33 b(solutions,)f(w)m(e)h(set)631 5313 y Fl(~)-64 b Ff(\034)688 5271 y Fx(\()p Fy(\006)p Fx(\))802 5313 y Fr(\()p Fo(\025)p Fr(\))28 b(:=)f Ff(\034)j Fr(\()p Fn(\006)p Fl(p)1352 5328 y Fy(?)1412 5313 y Fo(;)17 b(\025)p Fr(\))27 b(=)1682 5143 y Fk( )1761 5313 y Fn(\006)p Fl(p)1900 5328 y Fy(?)1959 5313 y Fo(;)2013 5246 y(m)2098 5210 y Fx(2)2160 5246 y Fr(+)22 b Fn(j)p Fl(p)2348 5261 y Fy(?)2407 5246 y Fn(j)2435 5198 y Fx(2)2496 5246 y Fn(\000)h Fo(\025)2653 5210 y Fx(2)p 2013 5290 680 4 v 2300 5382 a Fr(2)p Fo(\025)2702 5143 y Fk(!)3074 5313 y Fr(\(A11\))1653 5637 y(21)p eop %%Page: 22 22 22 21 bop 432 573 a Fr(and)32 b(\014nd)538 791 y(~)515 817 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))27 b(:=)h Fo(E)962 736 y Fk(\000)1015 817 y Fl(~)-63 b Ff(\034)1073 774 y Fx(\()p Fy(\006)p Fx(\))1187 817 y Fr(\()p Fo(\025)p Fr(\))1320 736 y Fk(\001)1393 817 y Fr(=)848 913 y Fk( )927 1084 y Fo(m)1012 1043 y Fx(2)1074 1084 y Fr(+)22 b Fn(j)p Fl(p)1262 1099 y Fy(?)1321 1084 y Fn(j)1349 1035 y Fx(2)1410 1084 y Fr(+)1518 1016 y(\()p Fo(m)1641 980 y Fx(2)1703 1016 y Fr(+)g Fn(j)p Fl(p)1891 1031 y Fy(?)1950 1016 y Fn(j)1978 968 y Fx(2)2039 1016 y Fn(\000)h Fo(\025)2196 980 y Fx(2)2235 1016 y Fr(\))2273 980 y Fx(2)p 1518 1061 795 4 v 1843 1152 a Fr(4)p Fo(\025)1949 1123 y Fx(2)2322 913 y Fk(!)2401 936 y Fx(1)p Fm(=)p Fx(2)2539 1084 y Fr(=)2653 1016 y Fo(m)2738 980 y Fx(2)2799 1016 y Fr(+)f Fn(j)p Fl(p)2987 1031 y Fy(?)3046 1016 y Fn(j)3074 968 y Fx(2)3136 1016 y Fr(+)g Fo(\025)3291 980 y Fx(2)p 2653 1061 678 4 v 2938 1152 a Fr(2)p Fo(\025)3423 1084 y Fr(\(A12\))432 1384 y(for)32 b Fo(\025)27 b(>)h Fr(0.)43 b(In)33 b(view)g(of)f(the)h (equalities)1565 1603 y Fl(~)-64 b Ff(\034)1622 1560 y Fx(\()p Fy(\006)p Fx(\))1736 1603 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))28 b(=)f(\()p Fn(\006)p Fl(p)2314 1618 y Fy(?)2374 1603 y Fo(;)17 b(p)2467 1618 y Fm(z)2507 1603 y Fr(\))878 b(\(A13\))432 1822 y(and)1523 1942 y Fo(\024)1596 1831 y Fk(\020)1663 1942 y Fl(~)-64 b Ff(\034)1721 1899 y Fx(\()p Fy(\006)p Fx(\))1835 1942 y Fr(\()p Fo(\025)p Fr(\))1968 1831 y Fk(\021)2055 1942 y Fr(=)27 b Fo(\025)h(;)44 b(\025)28 b(>)f Fr(0)p Fo(;)845 b Fr(\(A14\))432 2135 y(w)m(e)33 b(ma)m(y)g(write)1066 2370 y(~)1053 2396 y Fo(\036)1111 2411 y Fm(r)1149 2396 y Fr(\()p Fo(\017)1226 2411 y Fm(r)1264 2396 y Fl(p)1326 2411 y Fy(?)1385 2396 y Fo(;)17 b(z)t Fr(\))28 b(=)1648 2261 y Fk(Z)1703 2486 y Fe(R)1772 2396 y Fo(\036)1830 2411 y Fm(r)1884 2286 y Fk(\020)1952 2396 y Fl(~)-64 b Ff(\034)2009 2353 y Fx(\()p Fm(\017)2065 2361 y Fh(r)2100 2353 y Fx(\))2131 2396 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))2401 2286 y Fk(\021)2498 2370 y Fr(~)2478 2396 y Fo( )2541 2412 y Fm(\024)p Fx(\()p Fi(p)p Fx(\))p Fm(;r)2739 2396 y Fr(\()p Fo(z)t Fr(\))17 b Fo(dp)2981 2411 y Fm(z)3021 2396 y Fo(:)375 b Fr(\(A15\))432 2661 y(Here,)648 2853 y(~)628 2880 y Fo( )691 2895 y Fm(\025;r)790 2880 y Fr(\()p Fo(z)t Fr(\))84 b(:=)f Fo( )1253 2908 y Fi(~)-46 b Fa(\034)1295 2883 y Fg(\()p Fh(\017)1345 2891 y(r)1379 2883 y Fg(\))1407 2908 y Fx(\()p Fm(\025)p Fx(\))p Fm(;r)1561 2880 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))p Fo(e)1807 2839 y Fy(\000)p Fm(i)p Fi(p)1931 2851 y Fc(?)1983 2839 y Fy(\001)p Fi(r)2036 2856 y Fc(?)1012 3133 y Fr(=)1185 2962 y Fk( )1264 3133 y Fl(1)22 b Fr(+)1450 3065 y Fo(e)1500 3068 y(=)1495 3065 y(n)1565 3024 y Fr(~)1573 3056 y Fo(=)1553 3065 y(A)p Fr(\()p Fo(t)h Fn(\000)f Fo(z)t Fr(\))p 1450 3110 460 4 v 1588 3201 a(2)p Fo(\017)1676 3216 y Fm(r)1714 3201 y Fo(\025)1919 2962 y Fk(!)2021 3133 y Fr(~)-56 b Fo(u)2070 3148 y Fm(r)2108 3133 y Fr(\()p Fo(\025)p Fr(\))p Fo(e)2286 3092 y Fy(\000)p Fm(i\017)2394 3100 y Fh(r)2428 3092 y Fm(\025t)2515 3133 y Fr(exp)18 b([)p Fo(i\017)2780 3148 y Fm(r)2819 3133 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))6 b(~)-55 b Fo(\027)3149 3148 y Fm(t)p Fy(\000)p Fm(z)3270 3133 y Fr(\()p Fo(\025)p Fr(\)])16 b Fo(;)-50 b Fr(\(A16\))432 3433 y(where)1309 3588 y(~)-55 b Fo(\027)1351 3603 y Fm(\030)1389 3588 y Fr(\()p Fo(\025)p Fr(\))28 b(:=)1690 3520 y(1)p 1690 3565 49 4 v 1690 3656 a(2)1766 3447 y Fk(\022)1849 3520 y Fo(m)1934 3484 y Fx(2)1996 3520 y Fr(+)22 b Fo(\031)2149 3535 y Fy(?)2208 3520 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)2400 3535 y Fy(?)2459 3520 y Fr(\))2497 3484 y Fx(2)p 1849 3565 688 4 v 2164 3656 a Fo(\025)2568 3588 y Fn(\000)23 b Fo(\025)2725 3447 y Fk(\023)3423 3588 y Fr(\(A17\))432 3819 y(is)32 b(the)h(third)f(comp)s(onen)m(t)g(of)g Ff(\027)1616 3834 y Fm(\030)1686 3819 y Fr(de\014ned)i(in)e(\(39\).)43 b(F)-8 b(urthermore,)1568 4068 y(~)-55 b Fo(u)1618 4083 y Fm(r)1655 4068 y Fr(\()p Fo(\025)p Fr(\))28 b(:=)f Fo(u)2002 4083 y Fm(r)2056 3957 y Fk(\020)2124 4068 y Fl(~)-64 b Ff(\034)2181 4025 y Fx(\()p Fm(\017)2237 4033 y Fh(r)2272 4025 y Fx(\))2303 4068 y Fr(\()p Fo(\025)p Fr(\))2436 3957 y Fk(\021)2512 4068 y Fo(;)884 b Fr(\(A18\))432 4309 y(or,)32 b(more)g(explicitly)-8 b(,)1150 4624 y(~)-55 b Fo(u)1200 4639 y Fm(r)1237 4624 y Fr(\()p Fo(\025)p Fr(\))28 b(=)1501 4443 y Fk(s)p 1601 4443 437 4 v 1634 4531 a Fr(~)1611 4557 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))22 b(+)g Fo(m)p 1611 4601 417 4 v 1689 4704 a Fr(2)1761 4679 y(~)1738 4704 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))2054 4454 y Fk( )2133 4624 y Fl(1)22 b Fr(+)2319 4557 y Ff(\013)g Fn(\001)29 b Fl(~)-63 b Ff(\034)2532 4514 y Fx(\()p Fm(\017)2588 4522 y Fh(r)2623 4514 y Fx(\))2654 4557 y Fr(\()p Fo(\025)p Fr(\))p 2319 4601 469 4 v 2367 4679 a(~)2345 4704 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))22 b(+)g Fo(m)2797 4454 y Fk(!)2876 4753 y Fm(r)2930 4624 y Fo(;)466 b Fr(\(A19\))432 4929 y(where)48 b(the)f(index)h Fo(r)h Fr(on)e(the)g(righ)m(t-hand)f(side)h (denotes)i(the)e Fo(r)s Fr(th)g(column)f(of)g(the)432 5049 y(enclosed)33 b(matrix.)578 5169 y(The)h(functions)1219 5143 y(~)1199 5169 y Fo( )1262 5184 y Fm(\025;r)1394 5169 y Fr(ful\014ll)c(an)i(eigen)m(v)-5 b(alue)32 b(equation)h(closely) f(related)g(to)g(\(32\):)1584 5388 y(\004)1669 5362 y(~)1649 5388 y Fo( )1712 5403 y Fm(\025;r)1811 5388 y Fr(\()p Fo(z)t Fr(\))d(=)e Fo(\017)2107 5403 y Fm(r)2145 5388 y Fo(\025)2222 5362 y Fr(~)2202 5388 y Fo( )2265 5403 y Fm(\025;r)2364 5388 y Fr(\()p Fo(z)t Fr(\))p Fo(;)907 b Fr(\(A20\))2002 5637 y(22)p eop %%Page: 23 23 23 22 bop 83 573 a Fr(where)524 796 y(\004)83 b(:=)g Fo(\013)920 811 y Fm(x)964 796 y Fo(p)1013 811 y Fm(x)1079 796 y Fr(+)22 b Fo(\013)1239 811 y Fm(y)1280 796 y Fo(p)1329 811 y Fm(y)1393 796 y Fn(\000)g Fo(i\013)1587 811 y Fm(z)1627 796 y Fo(@)1678 811 y Fm(z)1741 796 y Fn(\000)h Fo(e)p Ff(\013)1976 772 y Fl(~)1962 796 y(A)o Fr(\()p Fo(t)f Fn(\000)h Fo(z)t Fr(\))g(+)f Fo(\014)6 b(m)22 b Fr(+)g Fo(i@)2799 811 y Fm(z)685 950 y Fr(=)97 b Fo(i)17 b Fr(\()p Fl(1)22 b Fn(\000)g Fo(\013)1185 965 y Fm(z)1225 950 y Fr(\))17 b Fo(@)1331 965 y Fm(z)1393 950 y Fr(+)22 b Fo(\013)1553 965 y Fm(x)1597 950 y Fo(p)1646 965 y Fm(x)1712 950 y Fr(+)g Fo(\013)1872 965 y Fm(y)1914 950 y Fo(p)1963 965 y Fm(y)2026 950 y Fn(\000)h Fo(e)p Ff(\013)2261 926 y Fl(~)2247 950 y(A)o Fr(\()p Fo(t)g Fn(\000)f Fo(z)t Fr(\))h(+)f Fo(\014)6 b(m)194 b Fr(\(A21\))83 1173 y(is)26 b(a)h(formally)d(self-adjoin)m(t)h(ordinary)h(di\013eren)m(tial)f(op)s (erator.)41 b(The)27 b(v)-5 b(alidit)m(y)25 b(of)33 b(\(A20\))83 1294 y(is)f(sho)m(wn)i(easily:)43 b(F)-8 b(rom)31 b(\(A16\))h(w)m(e)h (infer)f(that)1052 1517 y Fo( )1120 1545 y Fi(~)-46 b Fa(\034)1162 1520 y Fg(\()p Fh(\017)1212 1528 y(r)1247 1520 y Fg(\))1275 1545 y Fx(\()p Fm(\025)p Fx(\))p Fm(;r)1429 1517 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))27 b(=)h Fo(e)1806 1475 y Fm(i)p Fi(p)1875 1487 y Fc(?)1927 1475 y Fy(\001)p Fi(r)1980 1492 y Fc(?)2057 1490 y Fr(~)2037 1517 y Fo( )2100 1532 y Fm(\025;r)2199 1517 y Fr(\()p Fo(z)t Fr(\))p Fo(;)723 b Fr(\(A22\))83 1740 y(so)33 b(that)f(b)m(y)j(\(32\))o(,)418 1990 y(\()p Fo(H)8 b Fr(\()p Fo(t)p Fr(\))22 b(+)g Fo(i@)860 2005 y Fm(z)900 1990 y Fr(\))955 1879 y Fk(\020)1014 1990 y Fo(e)1059 1949 y Fm(i)p Fi(p)1128 1961 y Fc(?)1181 1949 y Fy(\001)p Fi(r)1234 1966 y Fc(?)1310 1964 y Fr(~)1290 1990 y Fo( )1353 2005 y Fm(\025;r)1452 1990 y Fr(\()p Fo(z)t Fr(\))1577 1879 y Fk(\021)1721 1990 y Fr(=)82 b Fo(\017)1918 2005 y Fm(r)1957 1990 y Fo(\024)p Fr(\()8 b Fl(~)-64 b Ff(\034)2116 1947 y Fx(\()p Fm(\017)2172 1955 y Fh(r)2207 1947 y Fx(\))2238 1990 y Fr(\()p Fo(\025)p Fr(\)\))p Fo( )2477 2019 y Fi(~)-46 b Fa(\034)2519 1993 y Fg(\()p Fh(\017)2569 2001 y(r)2604 1993 y Fg(\))2632 2019 y Fx(\()p Fm(\025)p Fx(\))p Fm(;r)2785 1990 y Fr(\()p Fl(r)p Fo(;)17 b(t)p Fr(\))1721 2180 y(=)82 b Fo(\017)1918 2195 y Fm(r)1957 2180 y Fo(\025)17 b(e)2076 2139 y Fm(i)p Fi(p)2145 2151 y Fc(?)2197 2139 y Fy(\001)p Fi(r)2250 2156 y Fc(?)2326 2154 y Fr(~)2306 2180 y Fo( )2369 2195 y Fm(\025;r)2469 2180 y Fr(\()p Fo(z)t Fr(\))p Fo(:)453 b Fr(\(A23\))83 2403 y(Since)251 2626 y(\()p Fn(\000)p Fo(i\013)461 2641 y Fm(x)506 2626 y Fo(@)557 2641 y Fm(x)623 2626 y Fn(\000)23 b Fo(i\013)818 2641 y Fm(y)860 2626 y Fo(@)911 2641 y Fm(y)952 2626 y Fr(\))1007 2516 y Fk(\020)1067 2626 y Fo(e)1112 2585 y Fm(i)p Fi(p)1181 2597 y Fc(?)1233 2585 y Fy(\001)p Fi(r)1286 2602 y Fc(?)1363 2600 y Fr(~)1343 2626 y Fo( )1406 2641 y Fm(\025;r)1505 2626 y Fr(\()p Fo(z)t Fr(\))1630 2516 y Fk(\021)1718 2626 y Fr(=)k(\()p Fo(\013)1921 2641 y Fm(x)1965 2626 y Fo(p)2014 2641 y Fm(x)2080 2626 y Fr(+)22 b Fo(\013)2240 2641 y Fm(y)2281 2626 y Fo(p)2330 2641 y Fm(y)2372 2626 y Fr(\))2426 2516 y Fk(\020)2486 2626 y Fo(e)2531 2585 y Fm(i)p Fi(p)2600 2597 y Fc(?)2652 2585 y Fy(\001)p Fi(r)2706 2602 y Fc(?)2782 2600 y Fr(~)2762 2626 y Fo( )2825 2641 y Fm(\025;r)2924 2626 y Fr(\()p Fo(z)t Fr(\))3049 2516 y Fk(\021)3126 2626 y Fo(;)3074 2774 y Fr(\(A24\))83 2894 y(equation)32 b(\(A20\))g(follo)m(ws)g(immediately)-8 b(.)229 3016 y(P)m(erforming)32 b(the)h(substitution)1304 3239 y Fo(\033)e Fr(:)d Fo(p)1494 3254 y Fm(z)1562 3239 y Fn(7!)f Fo(\024)p Fr(\()p Fl(p)p Fr(\))h Fn(\021)g Fo(\025;)974 b Fr(\(A25\))83 3462 y(w)m(e)34 b(are)e(led)g(to)232 3721 y(~)219 3747 y Fo(\036)277 3762 y Fm(r)315 3747 y Fr(\()p Fo(\017)392 3762 y Fm(r)430 3747 y Fl(p)492 3762 y Fy(?)551 3747 y Fo(;)17 b(z)t Fr(\))28 b(=)824 3680 y(1)p 824 3724 49 4 v 824 3815 a(2)899 3611 y Fk(Z)999 3638 y Fy(1)955 3837 y Fx(0)1090 3747 y Fo(\036)1148 3762 y Fm(r)1203 3636 y Fk(\020)1270 3747 y Fl(~)-64 b Ff(\034)1327 3704 y Fx(\()p Fm(\017)1383 3712 y Fh(r)1418 3704 y Fx(\))1450 3747 y Fr(\()p Fo(\025)p Fr(\))1583 3636 y Fk(\021)1678 3721 y Fr(~)1658 3747 y Fo( )1721 3762 y Fm(\025;r)1821 3747 y Fr(\()p Fo(z)t Fr(\))1963 3577 y Fk( )2052 3680 y Fo(m)2137 3644 y Fx(2)2199 3680 y Fr(+)22 b Fn(j)o Fl(p)2386 3695 y Fy(?)2446 3680 y Fn(j)2473 3631 y Fx(2)p 2052 3724 462 4 v 2234 3815 a Fo(\025)2291 3787 y Fx(2)2545 3747 y Fr(+)g(1)2692 3577 y Fk(!)2804 3747 y Fo(d\025;)135 b Fr(\(A26\))83 4052 y(where)34 b(w)m(e)f(ha)m(v)m(e)h(used)359 4270 y Fo(d)p 330 4314 108 4 v 330 4405 a(d\025)448 4337 y(\033)507 4296 y Fy(\000)p Fx(1)601 4337 y Fr(\()p Fo(\025)p Fr(\))28 b(=)904 4270 y Fo(d)p 875 4314 V 875 4405 a(d\025)1003 4270 y(m)1088 4233 y Fx(2)1150 4270 y Fr(+)22 b Fn(j)o Fl(p)1337 4285 y Fy(?)1397 4270 y Fn(j)1424 4221 y Fx(2)1486 4270 y Fn(\000)g Fo(\025)1642 4233 y Fx(2)p 1003 4314 680 4 v 1289 4405 a Fr(2)p Fo(\025)1719 4337 y Fr(=)28 b Fn(\000)1910 4270 y Fr(1)p 1910 4314 49 4 v 1910 4405 a(2)1986 4167 y Fk( )2075 4270 y Fo(m)2160 4233 y Fx(2)2221 4270 y Fr(+)22 b Fn(j)p Fl(p)2409 4285 y Fy(?)2468 4270 y Fn(j)2496 4221 y Fx(2)p 2075 4314 462 4 v 2257 4405 a Fo(\025)2314 4377 y Fx(2)2568 4337 y Fr(+)g(1)2715 4167 y Fk(!)2810 4337 y Fo(:)237 b Fr(\(A27\))229 4666 y(An)31 b(attempt)e(to)h(pro)m(v)m(e)h(\(A10\))e(with)1644 4640 y(~)1631 4666 y Fo(\036)1689 4681 y Fm(r)1757 4666 y Fr(from)g(\(A26\))g(directly)g(b)m(y)i(in)m(terc)m(hanging)83 4786 y(the)38 b(orders)h(of)f(in)m(tegration)e(fails)g(b)s(ecause)j (the)g(assumptions)f(of)f(F)-8 b(ubini's)37 b(theorem)83 4907 y(are)47 b(not)f(ful\014lled.)85 b(In)47 b(order)g(to)f(apply)g (the)i(theorem,)i(the)d(term)2754 4880 y(~)2734 4907 y Fo( )2797 4922 y Fm(\025;r)2897 4907 y Fr(\()p Fo(z)t Fr(\))g(w)m(ould)83 5027 y(ha)m(v)m(e)g(to)e(deca)m(y)h(for)f(large)f (v)-5 b(alues)46 b(of)e Fn(j)p Fo(z)t Fn(j)p Fr(.)82 b(As)46 b(w)m(e)g(shall)e(see,)50 b(w)m(e)c(can)g(a)m(v)m(oid)f(this)83 5147 y(problem)35 b(b)m(y)h(constructing)g(an)g(analytic)f(con)m(tin)m (uation)f(of)i(\(A16\))f(to)h(v)-5 b(alues)35 b Fo(\025)e Fn(2)h Fj(C)83 5268 y Fr(with)47 b(Re)16 b Fo(\025)53 b(>)f Fr(0,)e(and)d(then)h(in)e(a)h(limiting)42 b(pro)s(cedure,)52 b(the)c(desired)f(result)g(will)83 5388 y(follo)m(w.)71 b(F)-8 b(or)41 b(this)h(purp)s(ose)h(w)m(e)g(need)g(to)f(examine)g(the) g(Dirac)f(spinor)48 b(~)-55 b Fo(u)2913 5403 y Fm(r)2992 5388 y Fr(b)s(ecause)1653 5637 y(23)p eop %%Page: 24 24 24 23 bop 432 573 a Fr(its)32 b(de\014nition)g(\(A19\))g(migh)m(t)f (pro)s(duce)i(a)g(singularit)m(y)e(and)h(a)h(branc)m(h)g(cut)h(at)e (certain)432 693 y(v)-5 b(alues)32 b(of)g Fo(\025)p Fr(.)44 b(Ho)m(w)m(ev)m(er,)35 b(for)d Fo(\025)27 b Fn(2)h Fj(C)48 b Fn(n)22 b(f)p Fr(0)p Fn(g)p Fr(,)1284 1008 y(Re)1438 983 y(~)1416 1008 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))27 b(=)1758 838 y Fk( )1846 941 y Fo(m)1931 905 y Fx(2)1993 941 y Fr(+)22 b Fn(j)p Fl(p)2181 956 y Fy(?)2240 941 y Fn(j)2268 892 y Fx(2)p 1846 985 462 4 v 1977 1077 a Fr(2)p Fn(j)p Fo(\025)p Fn(j)2139 1048 y Fx(2)2340 1008 y Fr(+)2448 941 y(1)p 2448 985 49 4 v 2448 1077 a(2)2506 838 y Fk(!)2602 1008 y Fr(Re)16 b Fo(\025:)606 b Fr(\(A28\))432 1340 y(So)32 b(whenev)m(er)j(Re)17 b Fo(\025)28 b(>)f Fr(0,)33 b(w)m(e)g(also)f(ha)m(v)m(e)i(Re)2142 1315 y(~)2120 1340 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))27 b Fo(>)h Fr(0)k(and)1663 1639 y(Re)1828 1546 y(~)1805 1572 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))22 b(+)g Fo(m)p 1805 1616 417 4 v 1883 1719 a Fr(2)1955 1694 y(~)1932 1719 y Fo(E)6 b Fr(\()p Fo(\025)p Fr(\))2259 1639 y Fo(>)27 b Fr(0)p Fo(:)985 b Fr(\(A29\))432 1935 y(Consequen)m(tly)-8 b(,)28 b(w)m(e)e(ma)m(y)e (extend)i(the)f(de\014nition)f(\(A19\))g(also)g(to)g(complex)g(argumen) m(ts)432 2055 y Fo(\025)33 b Fr(with)f(Re)17 b Fo(\025)28 b(>)h Fr(0)k(if)e(w)m(e)k(c)m(ho)s(ose)f(the)f(branc)m(h)h(cut)f(of)g (the)h(square)g(ro)s(ot)e(to)h(lie)e(on)i(the)432 2176 y(negativ)m(e)45 b(real)g(axis.)82 b(Observing)45 b(that)h(b)s(oth)f (sides)h(of)52 b(\(A20\))44 b(are)i(analytic)e(with)432 2296 y(resp)s(ect)34 b(to)e Fo(\025)h Fr(and)g(iden)m(tical)e(for)i (real)f Fo(\025)p Fr(,)g(w)m(e)i(deduce)h(the)e(v)-5 b(alidit)m(y)31 b(of)h(this)h(equation)432 2416 y(for)f Fp(al)5 b(l)42 b Fr(complex)32 b Fo(\025)h Fr(with)f(Re)17 b Fo(\025)27 b(>)h Fr(0.)578 2539 y(Next)f(insp)s(ect)f(the)g(function) 31 b(~)-54 b Fo(\027)1722 2554 y Fm(\030)1786 2539 y Fr(for)25 b(complex)g(argumen)m(ts)h Fo(\026)i Fr(=)f Fo(\025)9 b Fr(+)g Fo(i\016)29 b Fr(with)d Fo(\025)h(>)h Fr(0)432 2660 y(and)k Fo(\016)g Fn(2)c Fj(R)5 b Fr(:)487 2926 y(~)-54 b Fo(\027)530 2941 y Fm(\030)567 2926 y Fr(\()p Fo(\025)r Fr(+)r Fo(i\016)t Fr(\))28 b(=)1001 2858 y Fo(\025)p 1001 2903 57 4 v 1005 2994 a Fr(2)1085 2785 y Fk(\022)1168 2858 y Fo(m)1253 2822 y Fx(2)1315 2858 y Fr(+)22 b Fo(\031)1468 2873 y Fy(?)1527 2858 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)1719 2873 y Fy(?)1778 2858 y Fr(\))1816 2822 y Fx(2)p 1168 2903 688 4 v 1360 2994 a Fo(\025)1417 2965 y Fx(2)1479 2994 y Fr(+)22 b Fo(\016)1624 2965 y Fx(2)1887 2926 y Fn(\000)h Fr(1)2036 2785 y Fk(\023)2111 2926 y Fn(\000)2200 2858 y Fo(i\016)p 2200 2903 81 4 v 2216 2994 a Fr(2)2307 2785 y Fk(\022)2390 2858 y Fo(m)2475 2822 y Fx(2)2537 2858 y Fr(+)f Fo(\031)2690 2873 y Fy(?)2750 2858 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)2942 2873 y Fy(?)3000 2858 y Fr(\))3038 2822 y Fx(2)p 2390 2903 688 4 v 2582 2994 a Fo(\025)2639 2965 y Fx(2)2701 2994 y Fr(+)22 b Fo(\016)2846 2965 y Fx(2)3109 2926 y Fr(+)g(1)3256 2785 y Fk(\023)3346 2926 y Fo(;)50 b Fr(\(A30\))432 3211 y(so)32 b(that)h(w)m(e)g(obtain)f(for)g(the)h(exp)s(onen)m(tial)f (in)g(\(A16\):)951 3354 y Fk(\014)951 3414 y(\014)1000 3438 y Fr(exp)18 b([)p Fo(i\017)1265 3453 y Fm(r)1304 3438 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))6 b(~)-55 b Fo(\027)1634 3453 y Fm(t)p Fy(\000)p Fm(z)1755 3438 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\)])2132 3354 y Fk(\014)2132 3414 y(\014)1034 3584 y Fr(=)82 b(exp)18 b([)p Fn(\000)p Fo(\017)1501 3599 y Fm(r)1540 3584 y Fr(\()p Fo(z)27 b Fn(\000)c Fo(t)p Fr(\))17 b(Im)k(~)-55 b Fo(\027)2020 3599 y Fm(t)p Fy(\000)p Fm(z)2141 3584 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\)])1034 3788 y(=)82 b(exp)1358 3648 y Fk(\024)1411 3788 y Fo(\017)1450 3803 y Fm(r)1488 3788 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))1798 3721 y Fo(\016)p 1797 3765 49 4 v 1797 3857 a Fr(2)1873 3648 y Fk(\022)1956 3721 y Fo(m)2041 3685 y Fx(2)2103 3721 y Fr(+)g Fo(\031)2256 3736 y Fy(?)2315 3721 y Fr(\()p Fo(t)g Fn(\000)h Fo(z)t(;)17 b Fl(p)2665 3736 y Fy(?)2725 3721 y Fr(\))2763 3685 y Fx(2)p 1956 3765 847 4 v 2228 3857 a Fo(\025)2285 3828 y Fx(2)2346 3857 y Fr(+)22 b Fo(\016)2491 3828 y Fx(2)2834 3788 y Fr(+)g(1)2981 3648 y Fk(\023\025)3124 3788 y Fo(:)272 b Fr(\(A31\))432 4068 y(In)33 b(addition,)734 4282 y Fo(@)p 706 4326 114 4 v 706 4418 a(@)5 b(\025)835 4349 y Fr(~)-55 b Fo(\027)877 4364 y Fm(\030)915 4349 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\))28 b(=)33 b(~)-54 b Fo(\027)1434 4308 y Fy(0)1428 4374 y Fm(\030)1466 4349 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\))28 b(=)f Fn(\000)2017 4282 y Fr(1)p 2017 4326 49 4 v 2017 4418 a(2)2093 4209 y Fk(\022)2176 4282 y Fo(m)2261 4246 y Fx(2)2323 4282 y Fr(+)22 b Fo(\031)2476 4297 y Fy(?)2535 4282 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)2727 4297 y Fy(?)2786 4282 y Fr(\))2824 4246 y Fx(2)p 2176 4326 688 4 v 2333 4418 a Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\))2666 4389 y Fx(2)2895 4349 y Fr(+)g(1)3042 4209 y Fk(\023)3132 4349 y Fo(;)264 b Fr(\(A32\))432 4629 y(for)32 b(whic)m(h)h(w)m(e)g (\014nd)531 4850 y Fk(\014)531 4909 y(\014)570 4934 y Fr(~)-54 b Fo(\027)619 4893 y Fy(0)613 4959 y Fm(\030)651 4934 y Fr(\()p Fo(\025)21 b Fr(+)i Fo(i\016)t Fr(\))984 4850 y Fk(\014)984 4909 y(\014)1045 4934 y Fr(=)1158 4776 y Fk(\014)1158 4836 y(\014)1191 4861 y Fo(m)1276 4824 y Fx(2)1338 4861 y Fr(+)f Fo(\031)1491 4876 y Fy(?)1551 4861 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)1743 4876 y Fy(?)1801 4861 y Fr(\))1839 4824 y Fx(2)1900 4861 y Fr(+)22 b(\()p Fo(\025)g Fr(+)g Fo(i\016)t Fr(\))2332 4812 y Fx(2)2371 4776 y Fk(\014)2371 4836 y(\014)p 1158 4911 1247 4 v 1572 5015 a Fr(2)17 b Fn(j)o Fo(\025)23 b Fr(+)f Fo(i\016)t Fn(j)1950 4966 y Fx(2)2442 4934 y Fr(=)1130 5081 y Fk(q)p 1230 5081 1557 4 v 123 x Fr([)p Fo(m)1342 5175 y Fx(2)1404 5204 y Fr(+)g Fo(\031)1557 5219 y Fy(?)1616 5204 y Fr(\()p Fo(\030)5 b(;)17 b Fl(p)1808 5219 y Fy(?)1867 5204 y Fr(\))1905 5175 y Fx(2)1966 5204 y Fr(+)22 b Fo(\025)2121 5175 y Fx(2)2183 5204 y Fn(\000)g Fo(\016)2329 5175 y Fx(2)2369 5204 y Fr(])2396 5156 y Fx(2)2457 5204 y Fr(+)h(4)p Fo(\025)2662 5175 y Fx(2)2701 5204 y Fo(\016)2748 5175 y Fx(2)p 1130 5272 1657 4 v 1737 5363 a Fr(2)17 b(\()o Fo(\025)1897 5334 y Fx(2)1958 5363 y Fr(+)22 b Fo(\016)2103 5334 y Fx(2)2143 5363 y Fr(\))2825 5295 y Fn(6)p Fr(=)27 b(0)h Fo(;)44 b(\025)28 b(>)f Fr(0)p Fo(:)83 b Fr(\(A33\))2002 5637 y(24)p eop %%Page: 25 25 25 24 bop 83 573 a Fr(So)33 b(for)f Fo(\025)27 b(>)h Fr(0,)288 897 y(\()p Fo(z)f Fn(\000)22 b Fo(t)p Fr(\))590 871 y(~)570 897 y Fo( )633 912 y Fm(\025)p Fx(+)p Fm(i\016)o(;r)841 897 y Fr(\()p Fo(z)t Fr(\))84 b(=)1209 727 y Fk( )1288 897 y Fl(1)22 b Fr(+)1474 830 y Fo(e)1524 833 y(=)1519 830 y(n)1589 789 y Fr(~)1597 821 y Fo(=)1577 830 y(A)p Fr(\()p Fo(t)h Fn(\000)f Fo(z)t Fr(\))p 1474 874 460 4 v 1474 966 a(2)p Fo(\017)1562 981 y Fm(r)1600 966 y Fr(\()p Fo(\025)g Fr(+)g Fo(i\016)t Fr(\))1943 727 y Fk(!)2045 897 y Fr(~)-56 b Fo(u)2094 912 y Fm(r)2132 897 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\))p Fo(e)2510 856 y Fy(\000)p Fm(i\017)2618 864 y Fh(r)2652 856 y Fx(\()p Fm(\025)p Fx(+)p Fm(i\016)r Fx(\))p Fm(t)1209 1192 y Fn(\002)1453 1124 y(\000)p Fo(i\017)1602 1139 y Fm(r)p 1296 1169 502 4 v 1302 1260 a Fr(~)-55 b Fo(\027)1350 1226 y Fy(0)1344 1282 y Fm(t)p Fy(\000)p Fm(z)1464 1260 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\))1846 1124 y Fo(@)p 1818 1169 114 4 v 1818 1260 a(@)5 b(\025)1958 1192 y Fr(exp)18 b([)p Fo(i\017)2223 1207 y Fm(r)2261 1192 y Fr(\()p Fo(z)27 b Fn(\000)c Fo(t)p Fr(\))6 b(~)-55 b Fo(\027)2592 1207 y Fm(t)p Fy(\000)p Fm(z)2712 1192 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\)])17 b Fo(:)3074 1397 y Fr(\(A34\))229 1647 y(No)m(w)34 b(de\014ne)198 1926 y(^)185 1952 y Fo(\036)243 1967 y Fm(r)n(;\016)330 1952 y Fr(\()p Fo(z)t Fr(\))28 b(:=)624 1885 y(1)p 624 1929 49 4 v 624 2021 a(2)699 1817 y Fk(Z)799 1843 y Fy(1)755 2042 y Fx(0)890 1952 y Fo(\036)948 1967 y Fm(r)1002 1842 y Fk(\020)1070 1952 y Fl(~)-64 b Ff(\034)1127 1909 y Fx(\()p Fm(\017)1183 1917 y Fh(r)1218 1909 y Fx(\))1250 1952 y Fr(\()p Fo(\025)p Fr(\))1383 1842 y Fk(\021)1478 1926 y Fr(~)1458 1952 y Fo( )1521 1968 y Fm(\025)p Fy(\000)p Fm(i\016)r Fy(\001)p Fx(sgn)r([)o Fm(\017)1846 1976 y Fh(r)1880 1968 y Fx(\()p Fm(z)s Fy(\000)p Fm(t)p Fx(\)])q Fm(;r)2128 1952 y Fr(\()p Fo(z)t Fr(\))2270 1782 y Fk( )2359 1885 y Fo(m)2444 1849 y Fx(2)2506 1885 y Fr(+)22 b Fn(j)p Fl(p)2694 1900 y Fy(?)2753 1885 y Fn(j)2781 1837 y Fx(2)p 2359 1929 462 4 v 2542 2021 a Fo(\025)2599 1992 y Fx(2)2853 1952 y Fr(+)g(1)3000 1782 y Fk(!)3111 1952 y Fo(d\025)3074 2160 y Fr(\(A35\))83 2280 y(for)34 b Fo(\016)h Fn(\025)d Fr(0.)49 b(This)34 b(in)m(tegral)f(exists)j(b)s(ecause)g(the)f(in)m (tegrand)f(is)g(con)m(tin)m(uous)h(and)g Fo(\036)3210 2295 y Fm(r)3271 2280 y Fn(\016)91 2401 y Fl(~)-64 b Ff(\034)148 2358 y Fx(\()p Fm(\017)204 2366 y Fh(r)239 2358 y Fx(\))300 2401 y Fr(is)29 b(compactly)g(supp)s(orted)i(in)e(\(0) p Fo(;)17 b Fn(1)p Fr(\).)41 b(Comparing)29 b(with)g(\(A26\),)h(w)m(e)h (recognize)83 2521 y(that)h(for)g Fo(\016)g Fr(=)c(0,)1354 2638 y(^)1342 2664 y Fo(\036)1400 2679 y Fm(r)n(;)p Fx(0)1516 2664 y Fr(=)1632 2638 y(~)1619 2664 y Fo(\036)1677 2679 y Fm(r)1732 2664 y Fr(\()p Fo(\017)1809 2679 y Fm(r)1847 2664 y Fl(p)p Fo(;)17 b Fn(\001)p Fr(\))f Fo(:)1012 b Fr(\(A36\))83 2865 y(The)35 b(sign)e(of)g(the)h(imaginary)d(part)i(in)g (\(A35\))g(w)m(as)h(c)m(hosen)h(suc)m(h)g(that)f(for)f Fo(\016)g(>)c Fr(0,)34 b(w)m(e)83 2985 y(observ)m(e)g(exp)s(onen)m (tial)e(deca)m(y)i(with)f Fo(z)t Fr(:)548 3135 y Fk(\014)548 3195 y(\014)548 3255 y(\014)581 3135 y(\014)581 3195 y(\014)581 3255 y(\014)634 3223 y Fr(~)614 3250 y Fo( )677 3265 y Fm(\025)p Fy(\000)p Fm(i\016)r Fy(\001)p Fx(sgn)q([)p Fm(\017)1002 3273 y Fh(r)1036 3265 y Fx(\()p Fm(z)s Fy(\000)p Fm(t)p Fx(\)])q Fm(;r)1284 3250 y Fr(\()p Fo(z)t Fr(\))1409 3135 y Fk(\014)1409 3195 y(\014)1409 3255 y(\014)1443 3135 y(\014)1443 3195 y(\014)1443 3255 y(\014)1504 3250 y Fn(\024)714 3352 y Fk(\022)787 3492 y Fl(1)22 b Fr(+)1435 3425 y Fn(j)p Fo(e)p Fn(j)p 973 3469 1024 4 v 973 3560 a Fr(2)17 b Fn(j)o Fo(\025)22 b Fn(\000)h Fo(i\016)j Fn(\001)c Fr(sgn)17 b([)p Fo(\017)1621 3575 y Fm(r)1660 3560 y Fr(\()p Fo(z)26 b Fn(\000)d Fo(t)p Fr(\)])p Fn(j)2024 3378 y Fk(\014)2024 3437 y(\014)2024 3497 y(\014)2057 3378 y(\014)2057 3437 y(\014)2057 3497 y(\014)2095 3495 y Fo(=)2090 3492 y(n)2160 3451 y Fr(~)2168 3483 y Fo(=)2148 3492 y(A)p Fr(\()p Fo(t)g Fn(\000)f Fo(z)t Fr(\))2503 3378 y Fk(\014)2503 3437 y(\014)2503 3497 y(\014)2537 3378 y(\014)2537 3437 y(\014)2537 3497 y(\014)2570 3352 y(\023)714 3711 y Fn(\002)17 b(j)o(j)6 b Fr(~)-55 b Fo(u)919 3726 y Fm(r)957 3711 y Fr(\()p Fo(\025)22 b Fn(\000)g Fo(i\016)27 b Fn(\001)21 b Fr(sgn)c([)q Fo(\017)1550 3726 y Fm(r)1588 3711 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\)])q Fn(j)o(j)17 b Fo(e)2015 3670 y Fy(\000)p Fm(\017)2099 3678 y Fh(r)2133 3670 y Fm(\016)r Fy(\001)p Fx(sgn[)p Fm(\017)2337 3678 y Fh(r)2371 3670 y Fx(\()p Fm(z)s Fy(\000)p Fm(t)p Fx(\)])q Fm(t)714 3915 y Fn(\002)g Fr(exp)973 3775 y Fk(\024)1026 3915 y Fn(\000)p Fo(\017)1142 3930 y Fm(r)1197 3915 y Fn(j)p Fo(z)27 b Fn(\000)c Fo(t)p Fn(j)1504 3848 y Fo(\016)p 1503 3893 49 4 v 1503 3984 a Fr(2)1578 3775 y Fk(\022)1661 3848 y Fo(m)1746 3812 y Fx(2)1808 3848 y Fr(+)f Fo(\031)1961 3863 y Fy(?)2021 3848 y Fr(\()p Fo(t)g Fn(\000)h Fo(z)t(;)17 b Fl(p)2371 3863 y Fy(?)2430 3848 y Fr(\))2468 3812 y Fx(2)p 1661 3893 847 4 v 1933 3984 a Fo(\025)1990 3955 y Fx(2)2052 3984 y Fr(+)22 b Fo(\016)2197 3955 y Fx(2)2540 3915 y Fr(+)g(1)2687 3775 y Fk(\023\025)2829 3915 y Fo(;)218 b Fr(\(A37\))83 4215 y(where)34 b(\(A31\))e(w)m(as)h(used.)45 b(F)-8 b(rom)31 b(this)h(w)m(e)i(obtain)d(that)758 4375 y Fk(\020)818 4485 y Fo(z)h Fn(7!)1043 4459 y Fr(~)1023 4485 y Fo( )1086 4501 y Fm(\025)p Fy(\000)p Fm(i\016)r Fy(\001)p Fx(sgn)q([)o Fm(\017)1410 4509 y Fh(r)1444 4501 y Fx(\()p Fm(z)s Fy(\000)p Fm(t)p Fx(\)])q Fm(;r)1693 4485 y Fr(\()p Fo(z)t Fr(\))1818 4375 y Fk(\021)1905 4485 y Fn(2)c Fo(L)2065 4444 y Fx(2)2105 4485 y Fr(\()p Fj(R)5 b Fr(\))2247 4444 y Fx(4)2320 4485 y Fo(;)45 b(\016)31 b(>)d Fr(0)p Fo(:)428 b Fr(\(A38\))83 4776 y(This)35 b(enables)g(us)g(to)g(insp)s(ect)g(the)g(deca)m(y)h(prop)s(erties)e(of) 2266 4750 y(^)2254 4776 y Fo(\036)2312 4791 y Fm(r)n(;\016)2433 4776 y Fr(for)h Fn(j)o Fo(z)27 b Fn(\000)c Fo(t)p Fn(j)31 b(!)g(1)p Fr(.)49 b(W)-8 b(e)83 4897 y(\014rst)33 b(see)h(that)1067 4925 y Fk(\014)1067 4985 y(\014)1067 5045 y(\014)1100 4925 y(\014)1100 4985 y(\014)1100 5045 y(\014)1146 5013 y Fr(^)1134 5040 y Fo(\036)1192 5055 y Fm(r)n(;\016)1279 5040 y Fr(\()p Fo(z)t Fr(\))1404 4925 y Fk(\014)1404 4985 y(\014)1404 5045 y(\014)1438 4925 y(\014)1438 4985 y(\014)1438 5045 y(\014)1498 5040 y Fn(\024)29 b Fo(C)1674 5055 y Fx(0)1713 5040 y Fr(\()p Fo(\036)p Fr(\))e Fo(;)44 b(\016)32 b Fn(2)c Fr([0)p Fo(;)17 b Fr(1])p Fo(;)737 b Fr(\(A39\))83 5268 y(for)29 b(a)f(constan)m(t)i Fo(C)766 5283 y Fx(0)806 5268 y Fr(\()p Fo(\036)p Fr(\).)41 b(This)30 b(is)e(clear)h(since)g(the)h(in)m(tegrand)f(in)f(\(A35\))h(is)f(con)m (tin)m(uous)83 5388 y(with)h(resp)s(ect)h(to)f Fo(\016)k Fr(as)c(w)m(ell)g(as)g Fo(\025)g Fr(and)g(again)f(since)i Fo(\036)2074 5403 y Fm(r)2126 5388 y Fn(\016)23 b Fl(~)-64 b Ff(\034)2257 5345 y Fx(\()p Fm(\017)2313 5353 y Fh(r)2347 5345 y Fx(\))2408 5388 y Fr(has)30 b(compact)e(supp)s(ort.)1653 5637 y(25)p eop %%Page: 26 26 26 25 bop 432 573 a Fr(F)-8 b(urthermore,)32 b(application)e(of)39 b(\(A34\))32 b(giv)m(es)569 853 y(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))864 827 y(^)851 853 y Fo(\036)909 868 y Fm(r)n(;\016)997 853 y Fr(\()p Fo(z)t Fr(\))83 b(=)1374 786 y(1)p 1374 830 49 4 v 1374 921 a(2)1449 717 y Fk(Z)1549 744 y Fy(1)1505 943 y Fx(0)1640 853 y Fo(\036)1698 868 y Fm(r)1753 742 y Fk(\020)1820 853 y Fl(~)-64 b Ff(\034)1877 810 y Fx(\()p Fm(\017)1933 818 y Fh(r)1968 810 y Fx(\))2000 853 y Fr(\()p Fo(\025)p Fr(\))2133 742 y Fk(\021)2209 683 y( )2297 786 y Fo(m)2382 750 y Fx(2)2444 786 y Fr(+)22 b Fn(j)p Fl(p)2632 801 y Fy(?)2691 786 y Fn(j)2719 737 y Fx(2)p 2297 830 462 4 v 2480 921 a Fo(\025)2537 893 y Fx(2)2791 853 y Fr(+)g(1)2938 683 y Fk(!)1364 1185 y Fn(\002)1458 1015 y Fk( )1537 1185 y Fl(1)g Fr(+)2054 1118 y Fo(e)2104 1121 y(=)2099 1118 y(n)2170 1077 y Fr(~)2177 1109 y Fo(=)2158 1118 y(A)p Fr(\()p Fo(t)g Fn(\000)h Fo(z)t Fr(\))p 1723 1162 1122 4 v 1723 1254 a(2)p Fo(\017)1811 1269 y Fm(r)1866 1254 y Fr(\()p Fo(\025)f Fn(\000)h Fo(i\016)j Fn(\001)c Fr(sgn)17 b([)p Fo(\017)2459 1269 y Fm(r)2497 1254 y Fr(\()p Fo(z)27 b Fn(\000)c Fo(t)p Fr(\)]\))2855 1015 y Fk(!)1364 1418 y Fn(\002)6 b Fr(~)-55 b Fo(u)1497 1433 y Fm(r)1535 1418 y Fr(\()p Fo(\025)22 b Fn(\000)h Fo(i\016)j Fn(\001)c Fr(sgn)17 b([)p Fo(\017)2128 1433 y Fm(r)2166 1418 y Fr(\()p Fo(z)27 b Fn(\000)c Fo(t)p Fr(\)]\))17 b(exp)g([)p Fn(\000)p Fo(i\017)2872 1433 y Fm(r)2911 1418 y Fo(\025t)p Fr(])1364 1609 y Fn(\002)g Fr(exp)1624 1528 y Fk(\010)1704 1609 y Fn(\000)23 b Fo(\017)1843 1624 y Fm(r)1881 1609 y Fo(\016)j Fn(\001)c Fr(sgn)17 b([)p Fo(\017)2224 1624 y Fm(r)2262 1609 y Fr(\()p Fo(z)27 b Fn(\000)c Fo(t)p Fr(\)])17 b Fo(t)2624 1528 y Fk(\011)2849 1541 y Fn(\000)p Fo(i\017)2998 1556 y Fm(r)p 2692 1586 502 4 v 2698 1677 a Fr(~)-55 b Fo(\027)2746 1643 y Fy(0)2740 1699 y Fm(t)p Fy(\000)p Fm(z)2860 1677 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\016)t Fr(\))1364 1876 y Fn(\002)1480 1809 y Fo(@)p 1451 1853 114 4 v 1451 1944 a(@)5 b(\025)1592 1876 y Fr(exp)1758 1795 y Fk(\002)1799 1876 y Fo(i\017)1871 1891 y Fm(r)1910 1876 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\))6 b(~)-55 b Fo(\027)2240 1891 y Fm(t)p Fy(\000)p Fm(z)2361 1876 y Fr(\()p Fo(\025)22 b Fn(\000)g Fo(i\016)27 b Fn(\001)21 b Fr(sgn)d([)p Fo(\017)2954 1891 y Fm(r)2992 1876 y Fr(\()p Fo(z)27 b Fn(\000)22 b Fo(t)p Fr(\)])q(\))3340 1795 y Fk(\003)3398 1876 y Fo(d\025:)3423 2052 y Fr(\(A40\))432 2245 y(P)m(erforming)31 b(an)i(in)m(tegration)d(b)m(y)k(parts,)f(w)m(e)g(notice)g(that)f(also) 1277 2460 y Fn(j)p Fo(z)26 b Fn(\000)d Fo(t)p Fn(j)1556 2346 y Fk(\014)1556 2406 y(\014)1556 2465 y(\014)1588 2346 y(\014)1588 2406 y(\014)1588 2465 y(\014)1634 2434 y Fr(^)1622 2460 y Fo(\036)1680 2475 y Fm(r)n(;\016)1767 2460 y Fr(\()p Fo(z)t Fr(\))1892 2346 y Fk(\014)1892 2406 y(\014)1892 2465 y(\014)1926 2346 y(\014)1926 2406 y(\014)1926 2465 y(\014)1986 2460 y Fn(\024)29 b Fo(C)2162 2475 y Fx(1)2201 2460 y Fr(\()p Fo(\036)p Fr(\))e Fo(;)45 b(\016)31 b Fn(2)d Fr([0)p Fo(;)17 b Fr(1])p Fo(:)598 b Fr(\(A41\))432 2675 y(Concludingly)-8 b(,)933 2803 y Fk(\014)933 2863 y(\014)933 2923 y(\014)966 2803 y(\014)966 2863 y(\014)966 2923 y(\014)1012 2891 y Fr(^)999 2918 y Fo(\036)1057 2933 y Fm(r)n(;\016)1144 2918 y Fr(\()p Fo(z)t Fr(\))1269 2803 y Fk(\014)1269 2863 y(\014)1269 2923 y(\014)1303 2803 y(\014)1303 2863 y(\014)1303 2923 y(\014)1364 2918 y Fn(\024)1479 2850 y Fo(C)1549 2865 y Fx(0)1588 2850 y Fr(\()p Fo(\036)p Fr(\))22 b(+)g Fo(C)1912 2865 y Fx(1)1951 2850 y Fr(\()p Fo(\036)p Fr(\))p 1479 2895 606 4 v 1566 2986 a(1)g(+)g Fn(j)p Fo(z)27 b Fn(\000)22 b Fo(t)p Fn(j)2122 2918 y(\021)2348 2850 y Fo(C)7 b Fr(\()p Fo(\036)p Fr(\))p 2237 2895 432 4 v 2237 2986 a(1)22 b(+)g Fn(j)p Fo(z)27 b Fn(\000)c Fo(t)p Fn(j)2706 2918 y Fo(;)44 b(\016)32 b Fn(2)c Fr([0)p Fo(;)17 b Fr(1])p Fo(;)254 b Fr(\(A42\))432 3187 y(so)48 b(that)f(the)h(functions)1425 3161 y(^)1412 3187 y Fo(\036)1470 3202 y Fm(r)n(;\016)1605 3187 y Fr(,)k Fo(\016)57 b Fn(2)d Fr([0)p Fo(;)17 b Fr(1],)51 b(are)d(uniformly)e(b)s(ounded)i(in)f Fo(L)3424 3151 y Fx(2)3464 3187 y Fr(.)89 b(In)432 3308 y(addition,)31 b(w)m(e)i(easily)f(see)i(b)m(y)f(dominated)f(con)m(v)m(ergence)j(that) 968 3501 y Fn(8)p Fo(z)e Fn(2)28 b Fj(R)66 b Fr(:)1417 3475 y(^)1405 3501 y Fo(\036)1463 3516 y Fm(r)n(;\016)1550 3501 y Fr(\()p Fo(z)t Fr(\))28 b Fn(!)1843 3475 y Fr(^)1830 3501 y Fo(\036)1888 3516 y Fm(r)n(;)p Fx(0)1977 3501 y Fr(\()p Fo(z)t Fr(\))g(=)2246 3475 y(~)2234 3501 y Fo(\036)2292 3516 y Fm(r)2329 3501 y Fr(\()p Fo(\017)2406 3516 y Fm(r)2445 3501 y Fl(p)2507 3516 y Fy(?)2566 3501 y Fo(;)17 b(z)t Fr(\))28 b Fo(;)44 b(\016)32 b Fn(!)27 b Fr(0)3047 3460 y Fx(+)3106 3501 y Fo(:)290 b Fr(\(A43\))432 3694 y(Therefore,)1315 3788 y(^)1303 3814 y Fo(\036)1361 3829 y Fm(r)n(;\016)1475 3814 y Fn(!)1615 3788 y Fr(~)1603 3814 y Fo(\036)1661 3829 y Fm(r)1698 3814 y Fr(\()p Fo(\017)1775 3829 y Fm(r)1814 3814 y Fl(p)1876 3829 y Fy(?)1935 3814 y Fo(;)17 b Fn(\001)p Fr(\))27 b Fo(;)44 b(\016)32 b Fn(!)27 b Fr(0)2394 3773 y Fx(+)2481 3814 y Fo(;)44 b Fr(in)32 b Fo(L)2732 3773 y Fx(2)2772 3814 y Fo(:)624 b Fr(\(A44\))578 3977 y(The)34 b(ab)s(o)m(v)m(e)f(considerations)f (lead)g(us)h(to)985 4059 y Fk(D)1059 4144 y Fr(~)1046 4170 y Fo(\036)1104 4185 y Fm(r)1142 4170 y Fr(\()p Fo(\017)1219 4185 y Fm(r)1257 4170 y Fl(p)1319 4185 y Fy(?)1378 4170 y Fo(;)17 b Fn(\001)p Fr(\))p Fo(;)1544 4144 y Fr(~)1532 4170 y Fo(\036)1590 4185 y Fm(r)1624 4166 y Fc(0)1650 4170 y Fr(\()p Fo(\017)1727 4185 y Fm(r)1761 4166 y Fc(0)1787 4170 y Fl(p)1849 4185 y Fy(?)1909 4170 y Fo(;)g Fn(\001)p Fr(\))2019 4059 y Fk(E)2079 4239 y Fm(L)2127 4220 y Fg(2)2193 4170 y Fr(=)55 b(lim)2297 4237 y Fm(\016)r Fy(!)p Fx(0)2436 4218 y Fg(+)2504 4059 y Fk(D)2577 4144 y Fr(^)2564 4170 y Fo(\036)2622 4185 y Fm(r)n(;\016)2710 4170 y Fo(;)2766 4144 y Fr(^)2754 4170 y Fo(\036)2812 4185 y Fm(r)2846 4166 y Fc(0)2867 4185 y Fm(;\016)2925 4059 y Fk(E)2986 4239 y Fm(L)3034 4220 y Fg(2)3089 4170 y Fo(;)307 b Fr(\(A45\))432 4398 y(whic)m(h)39 b(necessitates)i(the)e(ev)-5 b(aluation)37 b(of)h(the)i(righ)m(t-hand)d(side)i(for)g Fo(\016)j(>)c Fr(0.)62 b(Due)39 b(to)432 4518 y(the)34 b(deca)m(y)i(prop)s(ert)m(y)e (\(A37\))g(that)g(is)f(ful\014lled)f(for)i(p)s(ositiv)m(e)f Fo(\016)t Fr(,)i(w)m(e)g(ma)m(y)f(no)m(w)g(in)m(v)m(ok)m(e)432 4638 y(F)-8 b(ubini's)31 b(theorem:)774 4744 y Fk(D)848 4829 y Fr(^)835 4855 y Fo(\036)893 4870 y Fm(r)n(;\016)980 4855 y Fo(;)1036 4829 y Fr(^)1024 4855 y Fo(\036)1082 4870 y Fm(r)1116 4851 y Fc(0)1138 4870 y Fm(;\016)1195 4744 y Fk(E)1256 4924 y Fm(L)1304 4905 y Fg(2)1426 4855 y Fr(=)1595 4787 y(1)p 1595 4832 49 4 v 1595 4923 a(4)1670 4719 y Fk(Z)1725 4945 y Fe(R)1794 4719 y Fk(Z)1849 4945 y Fe(R)p 1918 4728 552 4 v 1918 4855 a Fo(\036)1976 4870 y Fm(r)2030 4744 y Fk(\020)2098 4855 y Fl(~)-64 b Ff(\034)2155 4812 y Fx(\()p Fm(\017)2211 4820 y Fh(r)2246 4812 y Fx(\))2277 4855 y Fr(\()p Fo(\025)p Fr(\))2410 4744 y Fk(\021)2486 4855 y Fo(\036)2544 4870 y Fm(r)2578 4851 y Fc(0)2621 4744 y Fk(\020)2688 4855 y Fl(~)h Ff(\034)2746 4812 y Fx(\()p Fm(\017)2802 4831 y Fh(r)2833 4817 y Fc(0)2859 4812 y Fx(\))2890 4855 y Fr(\()p Fo(\025)2985 4814 y Fy(0)3009 4855 y Fr(\))3047 4744 y Fk(\021)1585 5152 y Fn(\002)1679 4982 y Fk( )1768 5085 y Fo(m)1853 5049 y Fx(2)1915 5085 y Fr(+)22 b Fn(j)o Fl(p)2102 5100 y Fy(?)2162 5085 y Fn(j)2189 5037 y Fx(2)p 1768 5129 462 4 v 1950 5221 a Fo(\025)2007 5192 y Fx(2)2261 5152 y Fr(+)g(1)2408 4982 y Fk(!)16 b( )2592 5085 y Fo(m)2677 5049 y Fx(2)2739 5085 y Fr(+)22 b Fn(j)o Fl(p)2926 5100 y Fy(?)2986 5085 y Fn(j)3013 5037 y Fx(2)p 2592 5129 V 2765 5221 a Fo(\025)2822 5192 y Fy(0)p Fx(2)3085 5152 y Fr(+)g(1)3232 4982 y Fk(!)1585 5388 y Fn(\002)p Fo(J)1716 5403 y Fm(r)n(;r)1800 5384 y Fc(0)1822 5403 y Fm(;\016)1879 5388 y Fr(\()p Fo(\025;)17 b(\025)2075 5347 y Fy(0)2098 5388 y Fr(\))g Fo(d\025)g(d\025)2386 5347 y Fy(0)2408 5388 y Fo(;)988 b Fr(\(A46\))2002 5637 y(26)p eop %%Page: 27 27 27 26 bop 83 573 a Fr(where)290 820 y Fo(J)344 835 y Fm(r)n(;r)428 816 y Fc(0)449 835 y Fm(;\016)507 820 y Fr(\()p Fo(\025;)17 b(\025)703 779 y Fy(0)726 820 y Fr(\))83 b(:=)1033 684 y Fk(Z)1088 910 y Fe(R)1157 709 y Fk(D)1237 793 y Fr(~)1218 820 y Fo( )1281 835 y Fm(\025)p Fy(\000)p Fm(i\016)r Fy(\001)p Fx(sgn)q([)o Fm(\017)1605 843 y Fh(r)1639 835 y Fx(\()p Fm(z)s Fy(\000)p Fm(t)p Fx(\)])q Fm(;r)1887 820 y Fr(\()p Fo(z)t Fr(\))p Fo(;)2077 793 y Fr(~)2056 820 y Fo( )2119 835 y Fm(\025)2160 816 y Fc(0)2184 835 y Fy(\000)p Fm(i\016)r Fy(\001)p Fx(sgn[)p Fm(\017)2467 854 y Fh(r)2498 840 y Fc(0)2523 835 y Fx(\()p Fm(z)s Fy(\000)p Fm(t)p Fx(\)])q Fm(;r)2768 816 y Fc(0)2794 820 y Fr(\()p Fo(z)t Fr(\))2919 709 y Fk(E)3014 820 y Fo(dz)860 1093 y Fr(=)1033 957 y Fk(Z)1132 984 y Fm(t)1088 1183 y Fy(\0001)1234 982 y Fk(D)1315 1067 y Fr(~)1295 1093 y Fo( )1358 1108 y Fm(\025)p Fx(+)p Fm(i\017)1507 1116 y Fh(r)1542 1108 y Fm(\016)o(;r)1629 1093 y Fr(\()p Fo(z)t Fr(\))p Fo(;)1818 1067 y Fr(~)1798 1093 y Fo( )1861 1108 y Fm(\025)1902 1089 y Fc(0)1925 1108 y Fx(+)p Fm(i\017)2033 1127 y Fh(r)2064 1113 y Fc(0)2090 1108 y Fm(\016)o(;r)2174 1089 y Fc(0)2200 1093 y Fr(\()p Fo(z)t Fr(\))2325 982 y Fk(E)2419 1093 y Fo(dz)1033 1361 y Fr(+)1126 1225 y Fk(Z)1225 1252 y Fy(1)1180 1451 y Fm(t)1316 1250 y Fk(D)1397 1335 y Fr(~)1377 1361 y Fo( )1440 1376 y Fm(\025)p Fy(\000)p Fm(i\017)1589 1384 y Fh(r)1624 1376 y Fm(\016)o(;r)1711 1361 y Fr(\()p Fo(z)t Fr(\))p Fo(;)1900 1335 y Fr(~)1880 1361 y Fo( )1943 1376 y Fm(\025)1984 1357 y Fc(0)2007 1376 y Fy(\000)p Fm(i\017)2115 1395 y Fh(r)2146 1381 y Fc(0)2172 1376 y Fm(\016)o(;r)2256 1357 y Fc(0)2281 1361 y Fr(\()p Fo(z)t Fr(\))2406 1250 y Fk(E)2501 1361 y Fo(dz)859 1585 y Fn(\021)97 b Fo(J)1096 1534 y Fx(\()p Fy(\000)p Fx(\))1087 1613 y Fm(r)n(;r)1171 1594 y Fc(0)1192 1613 y Fm(;\016)1250 1585 y Fr(\()p Fo(\025;)17 b(\025)1446 1544 y Fy(0)1468 1585 y Fr(\))22 b(+)g Fo(J)1689 1534 y Fx(\(+\))1680 1613 y Fm(r)n(;r)1764 1594 y Fc(0)1786 1613 y Fm(;\016)1843 1585 y Fr(\()p Fo(\025;)17 b(\025)2039 1544 y Fy(0)2062 1585 y Fr(\))p Fo(:)947 b Fr(\(A47\))83 1808 y(These)28 b(in)m(tegrals)d(can)h(b)s(e)g(ev)-5 b(aluated)26 b(b)m(y)g(emplo)m(ying)f(the)h(eigen)m(v)-5 b(alue)26 b(equation)f(\(A20\))83 1929 y(as)33 b(follo)m(ws:)274 2054 y Fk(Z)373 2081 y Fm(t)329 2280 y Fy(\0001)475 2079 y Fk(D)556 2164 y Fr(~)536 2190 y Fo( )599 2205 y Fm(\025)p Fx(+)p Fm(i\017)748 2213 y Fh(r)783 2205 y Fm(\016)o(;r)870 2190 y Fr(\()p Fo(z)t Fr(\))p Fo(;)17 b Fr(\004)1124 2164 y(~)1104 2190 y Fo( )1167 2205 y Fm(\025)1208 2186 y Fc(0)1231 2205 y Fx(+)p Fm(i\017)1339 2224 y Fh(r)1370 2210 y Fc(0)1396 2205 y Fm(\016)o(;r)1480 2186 y Fc(0)1505 2190 y Fr(\()p Fo(z)t Fr(\))1630 2079 y Fk(E)1725 2190 y Fo(dz)88 b Fr(=)83 b Fo(\017)2107 2205 y Fm(r)2141 2186 y Fc(0)2184 2190 y Fr(\()p Fo(\025)2279 2149 y Fy(0)2324 2190 y Fr(+)22 b Fo(i\017)2494 2205 y Fm(r)2528 2186 y Fc(0)2555 2190 y Fo(\016)t Fr(\))17 b Fo(J)2720 2139 y Fx(\()p Fy(\000)p Fx(\))2711 2218 y Fm(r)n(;r)2795 2199 y Fc(0)2816 2218 y Fm(;\016)2874 2190 y Fr(\()p Fo(\025;)g(\025)3070 2149 y Fy(0)3092 2190 y Fr(\))1909 2421 y(=)83 b(\()p Fo(\017)2145 2436 y Fm(r)2179 2417 y Fc(0)2205 2421 y Fo(\025)2262 2380 y Fy(0)2308 2421 y Fr(+)22 b Fo(i\016)t Fr(\))16 b Fo(J)2603 2370 y Fx(\()p Fy(\000)p Fx(\))2594 2449 y Fm(r)n(;r)2678 2430 y Fc(0)2700 2449 y Fm(;\016)2757 2421 y Fr(\()p Fo(\025;)h(\025)2953 2380 y Fy(0)2976 2421 y Fr(\))60 b(\(A48\))83 2644 y(and)383 2770 y Fk(Z)482 2796 y Fm(t)438 2995 y Fy(\0001)584 2795 y Fk(D)645 2906 y Fr(\004)730 2879 y(~)710 2906 y Fo( )773 2921 y Fm(\025)p Fx(+)p Fm(i\017)922 2929 y Fh(r)957 2921 y Fm(\016)o(;r)1044 2906 y Fr(\()p Fo(z)t Fr(\))p Fo(;)1233 2879 y Fr(~)1213 2906 y Fo( )1276 2921 y Fm(\025)1317 2902 y Fc(0)1340 2921 y Fx(+)p Fm(i\017)1448 2940 y Fh(r)1479 2926 y Fc(0)1505 2921 y Fm(\016)o(;r)1589 2902 y Fc(0)1615 2906 y Fr(\()p Fo(z)t Fr(\))1740 2795 y Fk(E)1884 2906 y Fr(=)p 2043 2819 505 4 v 83 w Fo(\017)2082 2921 y Fm(r)2137 2906 y Fr(\()p Fo(\025)22 b Fr(+)g Fo(i\017)2424 2921 y Fm(r)2463 2906 y Fo(\016)t Fr(\))p Fo(J)2611 2855 y Fx(\()p Fy(\000)p Fx(\))2602 2934 y Fm(r)n(;r)2686 2915 y Fc(0)2707 2934 y Fm(;\016)2765 2906 y Fr(\()p Fo(\025;)17 b(\025)2961 2864 y Fy(0)2983 2906 y Fr(\))1884 3136 y(=)83 b(\()p Fo(\017)2120 3151 y Fm(r)2158 3136 y Fo(\025)22 b Fn(\000)h Fo(i\016)t Fr(\))17 b Fo(J)2535 3085 y Fx(\()p Fy(\000)p Fx(\))2526 3165 y Fm(r)n(;r)2610 3146 y Fc(0)2631 3165 y Fm(;\016)2689 3136 y Fr(\()p Fo(\025;)g(\025)2885 3095 y Fy(0)2908 3136 y Fr(\))p Fo(:)101 b Fr(\(A49\))83 3360 y(Remem)m(b)s(ering)28 b(the)i(formal)d(self-adjoin)m(tness)i(of)g (\004)h(and)g(p)s(erforming)d(an)j(in)m(tegration)83 3480 y(b)m(y)k(parts,)e(w)m(e)i(see)g(that)534 3625 y Fk(Z)633 3651 y Fm(t)589 3851 y Fy(\0001)735 3650 y Fk(D)816 3735 y Fr(~)796 3761 y Fo( )859 3776 y Fm(\025)p Fx(+)p Fm(i\017)1008 3784 y Fh(r)1043 3776 y Fm(\016)o(;r)1130 3761 y Fr(\()p Fo(z)t Fr(\))p Fo(;)17 b Fr(\004)1385 3735 y(~)1364 3761 y Fo( )1427 3776 y Fm(\025)1468 3757 y Fc(0)1492 3776 y Fx(+)p Fm(i\017)1600 3795 y Fh(r)1631 3781 y Fc(0)1656 3776 y Fm(\016)o(;r)1740 3757 y Fc(0)1766 3761 y Fr(\()p Fo(z)t Fr(\))1891 3650 y Fk(E)1985 3761 y Fo(dz)534 4041 y Fn(\000)628 3905 y Fk(Z)727 3932 y Fm(t)683 4131 y Fy(\0001)829 3930 y Fk(D)890 4041 y Fr(\004)975 4015 y(~)955 4041 y Fo( )1018 4056 y Fm(\025)p Fx(+)p Fm(i\017)1167 4064 y Fh(r)1202 4056 y Fm(\016)o(;r)1289 4041 y Fr(\()p Fo(z)t Fr(\))p Fo(;)1479 4015 y Fr(~)1458 4041 y Fo( )1521 4056 y Fm(\025)1562 4037 y Fc(0)1586 4056 y Fx(+)p Fm(i\017)1694 4075 y Fh(r)1725 4061 y Fc(0)1750 4056 y Fm(\016)o(;r)1834 4037 y Fc(0)1860 4041 y Fr(\()p Fo(z)t Fr(\))1985 3930 y Fk(E)2080 4041 y Fo(dz)617 4321 y Fr(=)776 4186 y Fk(Z)875 4212 y Fm(t)831 4411 y Fy(\0001)977 4211 y Fk(D)1058 4295 y Fr(~)1038 4321 y Fo( )1101 4336 y Fm(\025)p Fx(+)p Fm(i\017)1250 4344 y Fh(r)1285 4336 y Fm(\016)o(;r)1372 4321 y Fr(\()p Fo(z)t Fr(\))p Fo(;)g(i)g Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)1869 4336 y Fm(z)1909 4321 y Fr(\))16 b Fo(@)2014 4336 y Fm(z)2075 4295 y Fr(~)2055 4321 y Fo( )2118 4336 y Fm(\025)2159 4317 y Fc(0)2182 4336 y Fx(+)p Fm(i\017)2290 4355 y Fh(r)2321 4341 y Fc(0)2346 4336 y Fm(\016)o(;r)2430 4317 y Fc(0)2456 4321 y Fr(\()p Fo(z)t Fr(\))2581 4211 y Fk(E)2676 4321 y Fo(dz)776 4602 y Fn(\000)870 4466 y Fk(Z)969 4492 y Fm(t)925 4691 y Fy(\0001)1071 4491 y Fk(D)1132 4602 y Fo(i)h Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)1460 4617 y Fm(z)1500 4602 y Fr(\))16 b Fo(@)1605 4617 y Fm(z)1665 4575 y Fr(~)1645 4602 y Fo( )1708 4617 y Fm(\025)p Fx(+)p Fm(i\017)1857 4625 y Fh(r)1892 4617 y Fm(\016)o(;r)1980 4602 y Fr(\()p Fo(z)t Fr(\))p Fo(;)2169 4575 y Fr(~)2149 4602 y Fo( )2212 4617 y Fm(\025)2253 4598 y Fc(0)2276 4617 y Fx(+)p Fm(i\017)2384 4636 y Fh(r)2415 4622 y Fc(0)2440 4617 y Fm(\016)o(;r)2524 4598 y Fc(0)2550 4602 y Fr(\()p Fo(z)t Fr(\))2675 4491 y Fk(E)2770 4602 y Fo(dz)617 4846 y Fr(=)83 b Fo(i)826 4736 y Fk(D)906 4820 y Fr(~)886 4846 y Fo( )949 4861 y Fm(\025)p Fx(+)p Fm(i\017)1098 4869 y Fh(r)1133 4861 y Fm(\016)o(;r)1220 4846 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)1653 4861 y Fm(z)1693 4846 y Fr(\))1767 4820 y(~)1747 4846 y Fo( )1810 4861 y Fm(\025)1851 4842 y Fc(0)1874 4861 y Fx(+)p Fm(i\017)1982 4880 y Fh(r)2013 4866 y Fc(0)2039 4861 y Fm(\016)o(;r)2123 4842 y Fc(0)2149 4846 y Fr(\()p Fo(t)p Fr(\))2260 4736 y Fk(E)2337 4846 y Fo(:)710 b Fr(\(A50\))83 5091 y(F)-8 b(rom)31 b(the)i(last)f(three)h(equations)g(w)m(e)h(deriv)m(e)f(that) 146 5355 y Fo(J)209 5304 y Fx(\()p Fy(\000)p Fx(\))200 5383 y Fm(r)n(;r)284 5364 y Fc(0)306 5383 y Fm(;\016)363 5355 y Fr(\()p Fo(\025;)17 b(\025)559 5314 y Fy(0)582 5355 y Fr(\))27 b(=)1087 5287 y Fo(i)p 761 5332 686 4 v 761 5423 a(\017)800 5438 y Fm(r)834 5419 y Fc(0)861 5423 y Fo(\025)918 5394 y Fy(0)963 5423 y Fn(\000)c Fo(\017)1102 5438 y Fm(r)1140 5423 y Fo(\025)f Fr(+)g(2)p Fo(i\016)1473 5244 y Fk(D)1553 5328 y Fr(~)1534 5355 y Fo( )1597 5370 y Fm(\025)p Fx(+)p Fm(i\017)1746 5378 y Fh(r)1780 5370 y Fm(\016)o(;r)1868 5355 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()o Fl(1)22 b Fn(\000)h Fo(\013)2300 5370 y Fm(z)2340 5355 y Fr(\))2414 5328 y(~)2394 5355 y Fo( )2457 5370 y Fm(\025)2498 5351 y Fc(0)2521 5370 y Fx(+)p Fm(i\017)2629 5389 y Fh(r)2660 5375 y Fc(0)2686 5370 y Fm(\016)o(;r)2770 5351 y Fc(0)2796 5355 y Fr(\()p Fo(t)p Fr(\))2907 5244 y Fk(E)2984 5355 y Fo(:)63 b Fr(\(A51\))1653 5637 y(27)p eop %%Page: 28 28 28 27 bop 432 573 a Fr(In)33 b(a)f(similar)d(w)m(a)m(y)-8 b(,)34 b(one)f(ma)m(y)f(sho)m(w)i(that)560 804 y Fo(J)623 753 y Fx(\(+\))614 833 y Fm(r)n(;r)698 814 y Fc(0)719 833 y Fm(;\016)777 804 y Fr(\()p Fo(\025;)17 b(\025)973 763 y Fy(0)995 804 y Fr(\))28 b(=)f Fn(\000)1598 737 y Fo(i)p 1251 781 725 4 v 1251 873 a(\017)1290 888 y Fm(r)1324 869 y Fc(0)1352 873 y Fo(\025)1409 844 y Fy(0)1454 873 y Fn(\000)c Fo(\017)1593 888 y Fm(r)1631 873 y Fo(\025)1688 888 y Fm(r)1748 873 y Fn(\000)g Fr(2)p Fo(i\016)2003 694 y Fk(D)2084 778 y Fr(~)2064 804 y Fo( )2127 819 y Fm(\025)p Fx(+)p Fm(i\017)2276 827 y Fh(r)2311 819 y Fm(\016)o(;r)2398 804 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)2831 819 y Fm(z)2870 804 y Fr(\))2945 778 y(~)2925 804 y Fo( )2988 819 y Fm(\025)3029 800 y Fc(0)3052 819 y Fx(+)p Fm(i\017)3160 838 y Fh(r)3191 824 y Fc(0)3216 819 y Fm(\016)o(;r)3300 800 y Fc(0)3326 804 y Fr(\()p Fo(t)p Fr(\))3437 694 y Fk(E)3515 804 y Fo(;)3423 971 y Fr(\(A52\))432 1091 y(so)32 b(that)739 1283 y Fo(J)793 1298 y Fm(r)n(;r)877 1279 y Fc(0)898 1298 y Fm(;\016)956 1283 y Fr(\()p Fo(\025;)17 b(\025)1152 1241 y Fy(0)1175 1283 y Fr(\))822 1488 y(=)981 1348 y Fk(\022)1409 1421 y Fo(i)p 1064 1465 724 4 v 1064 1556 a(\017)1103 1571 y Fm(r)1137 1552 y Fc(0)1164 1556 y Fo(\025)1221 1528 y Fy(0)1266 1556 y Fn(\000)23 b Fo(\017)1405 1571 y Fm(r)1443 1556 y Fo(\025)1500 1571 y Fm(r)1560 1556 y Fr(+)f(2)p Fo(i\016)1819 1488 y Fn(\000)2274 1421 y Fo(i)p 1929 1465 725 4 v 1929 1556 a(\017)1968 1571 y Fm(r)2002 1552 y Fc(0)2028 1556 y Fo(\025)2085 1528 y Fy(0)2131 1556 y Fn(\000)g Fo(\017)2269 1571 y Fm(r)2308 1556 y Fo(\025)2365 1571 y Fm(r)2425 1556 y Fn(\000)g Fr(2)p Fo(i\016)2663 1348 y Fk(\023)981 1731 y Fn(\002)1075 1620 y Fk(D)1156 1704 y Fr(~)1136 1731 y Fo( )1199 1746 y Fm(\025)p Fx(+)p Fm(i\017)1348 1754 y Fh(r)1382 1746 y Fm(\016)o(;r)1470 1731 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()o Fl(1)23 b Fn(\000)f Fo(\013)1902 1746 y Fm(z)1942 1731 y Fr(\))2017 1704 y(~)1997 1731 y Fo( )2060 1746 y Fm(\025)2101 1727 y Fc(0)2124 1746 y Fx(+)p Fm(i\017)2232 1765 y Fh(r)2263 1751 y Fc(0)2288 1746 y Fm(\016)o(;r)2372 1727 y Fc(0)2398 1731 y Fr(\()p Fo(t)p Fr(\))2509 1620 y Fk(E)822 1965 y Fr(=)1346 1898 y(4)p Fo(\016)p 991 1942 807 4 v 991 2045 a Fr(\()p Fo(\017)1068 2060 y Fm(r)1106 2045 y Fo(\025)g Fn(\000)h Fo(\017)1324 2060 y Fm(r)1358 2041 y Fc(0)1384 2045 y Fo(\025)1441 2017 y Fy(0)1465 2045 y Fr(\))1503 1997 y Fx(2)1564 2045 y Fr(+)f(4)p Fo(\016)1758 2017 y Fx(2)1824 1854 y Fk(D)1905 1939 y Fr(~)1885 1965 y Fo( )1948 1980 y Fm(\025)p Fx(+)p Fm(i\017)2097 1988 y Fh(r)2131 1980 y Fm(\016)o(;r)2219 1965 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()o Fl(1)23 b Fn(\000)f Fo(\013)2651 1980 y Fm(z)2691 1965 y Fr(\))2766 1939 y(~)2746 1965 y Fo( )2809 1980 y Fm(\025)2850 1961 y Fc(0)2873 1980 y Fx(+)p Fm(i\017)2981 1999 y Fh(r)3012 1985 y Fc(0)3037 1980 y Fm(\016)o(;r)3121 1961 y Fc(0)3147 1965 y Fr(\()p Fo(t)p Fr(\))3258 1854 y Fk(E)3335 1965 y Fo(:)61 b Fr(\(A53\))432 2219 y(This)32 b(leads)h(to)497 2326 y Fk(D)571 2410 y Fr(^)558 2436 y Fo(\036)616 2451 y Fm(r)n(;\016)703 2436 y Fo(;)760 2410 y Fr(^)747 2436 y Fo(\036)805 2451 y Fm(r)839 2432 y Fc(0)861 2451 y Fm(;\016)918 2326 y Fk(E)979 2505 y Fm(L)1027 2486 y Fg(2)1149 2436 y Fr(=)1318 2369 y Fo(\031)p 1318 2413 59 4 v 1323 2505 a Fr(2)1403 2301 y Fk(Z)1503 2327 y Fy(1)1459 2526 y Fx(0)1594 2301 y Fk(Z)1694 2327 y Fy(1)1650 2526 y Fx(0)1785 2436 y Fo(\021)1833 2451 y Fm(\016)1871 2436 y Fr(\()p Fo(\017)1948 2451 y Fm(r)1986 2436 y Fo(\025)23 b Fn(\000)f Fo(\017)2204 2451 y Fm(r)2238 2432 y Fc(0)2265 2436 y Fo(\025)2322 2395 y Fy(0)2345 2436 y Fr(\))p 2400 2310 552 4 v 17 w Fo(\036)2458 2451 y Fm(r)2512 2326 y Fk(\020)2579 2436 y Fl(~)-64 b Ff(\034)2637 2393 y Fx(\()p Fm(\017)2693 2401 y Fh(r)2727 2393 y Fx(\))2759 2436 y Fr(\()p Fo(\025)p Fr(\))2892 2326 y Fk(\021)2968 2436 y Fo(\036)3026 2451 y Fm(r)3060 2432 y Fc(0)3102 2326 y Fk(\020)3170 2436 y Fl(~)g Ff(\034)3227 2393 y Fx(\()p Fm(\017)3283 2412 y Fh(r)3314 2398 y Fc(0)3340 2393 y Fx(\))3372 2436 y Fr(\()p Fo(\025)3467 2395 y Fy(0)3490 2436 y Fr(\))3528 2326 y Fk(\021)1308 2734 y Fn(\002)1402 2563 y Fk( )1491 2666 y Fo(m)1576 2630 y Fx(2)1638 2666 y Fr(+)22 b Fn(j)o Fl(p)1825 2681 y Fy(?)1885 2666 y Fn(j)1912 2618 y Fx(2)p 1491 2711 462 4 v 1673 2802 a Fo(\025)1730 2773 y Fx(2)1984 2734 y Fr(+)g(1)2131 2563 y Fk(!)16 b( )2315 2666 y Fo(m)2400 2630 y Fx(2)2462 2666 y Fr(+)22 b Fn(j)p Fl(p)2650 2681 y Fy(?)2709 2666 y Fn(j)2737 2618 y Fx(2)p 2315 2711 V 2488 2802 a Fo(\025)2545 2773 y Fy(0)p Fx(2)2808 2734 y Fr(+)g(1)2955 2563 y Fk(!)1308 3006 y Fn(\002)1402 2895 y Fk(D)1483 2980 y Fr(~)1463 3006 y Fo( )1526 3021 y Fm(\025)p Fx(+)p Fm(i\017)1675 3029 y Fh(r)1710 3021 y Fm(\016)o(;r)1797 3006 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()p Fl(1)22 b Fn(\000)g Fo(\013)2229 3021 y Fm(z)2269 3006 y Fr(\))2344 2980 y(~)2324 3006 y Fo( )2387 3021 y Fm(\025)2428 3002 y Fc(0)2451 3021 y Fx(+)p Fm(i\017)2559 3040 y Fh(r)2590 3026 y Fc(0)2615 3021 y Fm(\016)o(;r)2699 3002 y Fc(0)2725 3006 y Fr(\()p Fo(t)p Fr(\))2836 2895 y Fk(E)2930 3006 y Fo(d\025)17 b(d\025)3163 2965 y Fy(0)3423 3006 y Fr(\(A54\))432 3219 y(with)1625 3364 y Fo(\021)1673 3379 y Fm(\016)1711 3364 y Fr(\()p Fo(x)p Fr(\))28 b(:=)2015 3297 y(2)p 2010 3341 59 4 v 2010 3433 a Fo(\031)2241 3297 y(\016)p 2089 3341 351 4 v 2089 3433 a(x)2144 3404 y Fx(2)2206 3433 y Fr(+)22 b(4)p Fo(\016)2400 3404 y Fx(2)2449 3364 y Fo(:)947 b Fr(\(A55\))578 3565 y(The)34 b(in)m(tegral)d(in)g(\(A54\))h(is)g(of)h(the)g(form)704 3697 y Fk(D)777 3781 y Fr(^)765 3807 y Fo(\036)823 3822 y Fm(r)n(;\016)910 3807 y Fo(;)966 3781 y Fr(^)954 3807 y Fo(\036)1012 3822 y Fm(r)1046 3803 y Fc(0)1067 3822 y Fm(;\016)1125 3697 y Fk(E)1186 3876 y Fm(L)1234 3857 y Fg(2)1300 3807 y Fr(=)1414 3740 y Fo(\031)p 1414 3784 59 4 v 1419 3876 a Fr(2)1499 3672 y Fk(Z)1599 3698 y Fy(1)1554 3897 y Fx(0)1690 3672 y Fk(Z)1790 3698 y Fy(1)1745 3897 y Fx(0)1881 3807 y Fo(\021)1929 3822 y Fm(\016)1967 3807 y Fr(\()p Fo(\017)2044 3822 y Fm(r)2082 3807 y Fo(\025)22 b Fn(\000)h Fo(\017)2300 3822 y Fm(r)2334 3803 y Fc(0)2361 3807 y Fo(\025)2418 3766 y Fy(0)2441 3807 y Fr(\))p Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(\025;)g(\025)2819 3766 y Fy(0)2841 3807 y Fr(\))g Fo(d\025)g(d\025)3129 3766 y Fy(0)3423 3807 y Fr(\(A56\))432 4050 y(with)38 b(a)h(function)f Fo(f)50 b Fr(dep)s(ending)39 b(con)m(tin)m(uously)g(on)g Fo(\016)t Fr(,)h Fo(\025)p Fr(,)h(and)e Fo(\025)2916 4014 y Fy(0)2939 4050 y Fr(.)63 b(The)39 b(restriction)442 4170 y Fo(f)11 b Fn(j)528 4199 y Fx([0)p Fm(;)p Fx(1])p Fy(\002)p Fx([0)p Fm(;)p Fy(1)p Fx(\))p Fy(\002)p Fx([0)p Fm(;)p Fy(1)p Fx(\))1147 4170 y Fr(has)32 b(compact)f(supp)s(ort)g(in)f ([0)p Fo(;)17 b Fr(1])i Fn(\002)h Fr(\(0)p Fo(;)d Fn(1)p Fr(\))h Fn(\002)h Fr(\(0)p Fo(;)e Fn(1)p Fr(\).)42 b(The)32 b(trivial)432 4303 y(extension)h(to)f([0)p Fo(;)17 b Fr(1])22 b Fn(\002)h Fj(R)1365 4267 y Fx(2)1411 4303 y Fr(,)33 b(i.)16 b(e.)1262 4555 y(~)1241 4582 y Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(\025;)g(\025)1581 4541 y Fy(0)1603 4582 y Fr(\))28 b(:=)1800 4411 y Fk(\()1880 4513 y Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(\025;)g(\025)2220 4477 y Fy(0)2243 4513 y Fr(\))97 b Fo(;)17 b(\025;)g(\025)2580 4477 y Fy(0)2630 4513 y Fo(>)28 b Fr(0)p Fo(;)1880 4658 y Fr(0)449 b Fo(;)17 b Fr(otherwise)p Fo(;)3423 4582 y Fr(\(A57\))432 4855 y(is)32 b(therefore)h(con)m(tin)m(uous,)g(and)g(w)m(e)h(ma)m(y)e (write)825 4979 y Fk(D)898 5063 y Fr(^)886 5089 y Fo(\036)944 5104 y Fm(r)n(;\016)1031 5089 y Fo(;)1087 5063 y Fr(^)1075 5089 y Fo(\036)1133 5104 y Fm(r)1167 5085 y Fc(0)1189 5104 y Fm(;\016)1246 4979 y Fk(E)1307 5158 y Fm(L)1355 5139 y Fg(2)1477 5089 y Fr(=)1646 5022 y Fo(\031)p 1646 5066 V 1651 5158 a Fr(2)1731 4954 y Fk(Z)1786 5179 y Fe(R)1855 4954 y Fk(Z)1910 5179 y Fe(R)1979 5089 y Fo(\021)2027 5104 y Fm(\016)2065 5089 y Fr(\()p Fo(\017)2142 5104 y Fm(r)2180 5089 y Fo(\025)22 b Fn(\000)h Fo(\017)2398 5104 y Fm(r)2432 5085 y Fc(0)2459 5089 y Fo(\025)2516 5048 y Fy(0)2539 5089 y Fr(\))2598 5063 y(~)2577 5089 y Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(\025;)g(\025)2917 5048 y Fy(0)2940 5089 y Fr(\))g Fo(d\025)g(d\025)3228 5048 y Fy(0)1477 5348 y Fr(=)1646 5281 y Fo(\031)p 1646 5325 V 1651 5416 a Fr(2)1731 5212 y Fk(Z)1786 5438 y Fe(R)1855 5212 y Fk(Z)1910 5438 y Fe(R)1979 5348 y Fo(\021)2027 5363 y Fm(\016)2065 5348 y Fr(\()p Fo(\025)22 b Fn(\000)h Fo(\025)2339 5307 y Fy(0)2362 5348 y Fr(\))2421 5322 y(~)2400 5348 y Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(\017)2621 5363 y Fm(r)2659 5348 y Fo(\025;)g(\017)2799 5363 y Fm(r)2833 5344 y Fc(0)2859 5348 y Fo(\025)2916 5307 y Fy(0)2940 5348 y Fr(\))g Fo(d\025)g(d\025)3228 5307 y Fy(0)3249 5348 y Fo(:)147 b Fr(\(A58\))2002 5637 y(28)p eop %%Page: 29 29 29 28 bop 83 573 a Fr(W)-8 b(e)31 b(are)g(required)g(to)f(determine)g (the)h(limit)c(of)k(\(A54\))f(for)g Fo(\016)h Fn(!)d Fr(0)2542 537 y Fx(+)2601 573 y Fr(,)j(see)g(\(A45\).)43 b(This)83 693 y(can)33 b(b)s(e)g(accomplished)e(with)i(the)g(help)f(of) g(Lemma)f(9)h(b)s(elo)m(w.)44 b(The)33 b(result)g(is)810 1092 y(lim)782 1160 y Fm(\016)r Fy(!)p Fx(0)921 1141 y Fg(+)989 982 y Fk(D)1063 1066 y Fr(^)1050 1092 y Fo(\036)1108 1107 y Fm(r)n(;\016)1195 1092 y Fo(;)1252 1066 y Fr(^)1239 1092 y Fo(\036)1297 1107 y Fm(r)1331 1089 y Fc(0)1353 1107 y Fm(;\016)1410 982 y Fk(E)1471 1161 y Fm(L)1519 1142 y Fg(2)1586 1092 y Fr(=)1699 1025 y Fo(\031)p 1699 1070 59 4 v 1704 1161 a Fr(2)1785 957 y Fk(Z)1840 1182 y Fe(R)1930 1066 y Fr(~)1909 1092 y Fo(f)10 b Fr(\(0)p Fo(;)17 b(\017)2137 1107 y Fm(r)2175 1092 y Fo(\025;)g(\017)2315 1107 y Fm(r)2349 1089 y Fc(0)2375 1092 y Fo(\025)p Fr(\))g Fo(d\025:)452 b Fr(\(A59\))83 1501 y(Owing)32 b(to)g(\(A45\))g(and)h (\(A57\))o(,)g(w)m(e)h(ha)m(v)m(e)445 1775 y Fk(D)519 1859 y Fr(~)506 1885 y Fo(\036)564 1900 y Fm(r)602 1885 y Fr(\()p Fo(\017)679 1900 y Fm(r)717 1885 y Fl(p)779 1900 y Fy(?)838 1885 y Fo(;)17 b Fn(\001)p Fr(\))p Fo(;)1004 1859 y Fr(~)992 1885 y Fo(\036)1050 1900 y Fm(r)1084 1881 y Fc(0)1109 1885 y Fr(\()p Fo(\017)1186 1900 y Fm(r)1220 1881 y Fc(0)1247 1885 y Fl(p)1309 1900 y Fy(?)1368 1885 y Fo(;)g Fn(\001)p Fr(\))1478 1775 y Fk(E)1539 1954 y Fm(L)1587 1935 y Fg(2)1653 1885 y Fr(=)55 b(lim)1757 1952 y Fm(\016)r Fy(!)p Fx(0)1896 1934 y Fg(+)1963 1775 y Fk(D)2037 1859 y Fr(^)2024 1885 y Fo(\036)2082 1900 y Fm(r)n(;\016)2169 1885 y Fo(;)2226 1859 y Fr(^)2213 1885 y Fo(\036)2271 1900 y Fm(r)2305 1881 y Fc(0)2327 1900 y Fm(;\016)2385 1775 y Fk(E)2445 1954 y Fm(L)2493 1935 y Fg(2)2560 1885 y Fr(=)27 b(0)362 b(\(A60\))83 2281 y(for)32 b Fo(\017)271 2296 y Fm(r)337 2281 y Fn(6)p Fr(=)c Fo(\017)480 2296 y Fm(r)514 2277 y Fc(0)540 2281 y Fr(,)33 b(whic)m(h)g(v)-5 b(alidates)31 b(\(A10\))h(in)g(this)g (case.)229 2447 y(No)m(w)i(assume)f Fo(\017)828 2462 y Fm(r)894 2447 y Fr(=)27 b Fo(\017)1036 2462 y Fm(r)1070 2443 y Fc(0)1125 2447 y Fr(=:)g Fo(\017)p Fr(.)44 b(Then)34 b(again)d(b)m(y)k(\(A45\))d(and)g(Lemma)g(9,)387 2720 y Fk(D)460 2805 y Fr(~)448 2831 y Fo(\036)506 2846 y Fm(r)544 2831 y Fr(\()p Fo(\017)621 2846 y Fm(r)659 2831 y Fl(p)721 2846 y Fy(?)780 2831 y Fo(;)17 b Fn(\001)p Fr(\))p Fo(;)946 2805 y Fr(~)934 2831 y Fo(\036)992 2846 y Fm(r)1026 2827 y Fc(0)1051 2831 y Fr(\()p Fo(\017)1128 2846 y Fm(r)1162 2827 y Fc(0)1189 2831 y Fl(p)1251 2846 y Fy(?)1310 2831 y Fo(;)g Fn(\001)p Fr(\))1420 2720 y Fk(E)1480 2900 y Fm(L)1528 2881 y Fg(2)470 3124 y Fr(=)639 3057 y Fo(\031)p 639 3101 V 644 3192 a Fr(2)724 2988 y Fk(Z)824 3015 y Fy(1)780 3214 y Fx(0)p 915 2997 517 4 v 915 3124 a Fo(\036)973 3139 y Fm(r)1028 3013 y Fk(\020)1095 3124 y Fl(~)-64 b Ff(\034)1152 3081 y Fx(\()p Fm(\017)p Fx(\))1240 3124 y Fr(\()p Fo(\025)p Fr(\))1373 3013 y Fk(\021)1449 3124 y Fo(\036)1507 3139 y Fm(r)1541 3120 y Fc(0)1583 3013 y Fk(\020)1651 3124 y Fl(~)g Ff(\034)1708 3081 y Fx(\()p Fm(\017)p Fx(\))1796 3124 y Fr(\()p Fo(\025)p Fr(\))1929 3013 y Fk(\021)2005 2954 y( )2094 3057 y Fo(m)2179 3020 y Fx(2)2240 3057 y Fr(+)23 b Fn(j)o Fl(p)2428 3072 y Fy(?)2487 3057 y Fn(j)2515 3008 y Fx(2)p 2094 3101 462 4 v 2276 3192 a Fo(\025)2333 3164 y Fx(2)2587 3124 y Fr(+)f(1)2734 2954 y Fk(!)2812 2976 y Fx(2)629 3396 y Fn(\002)723 3286 y Fk(D)804 3370 y Fr(~)784 3396 y Fo( )847 3411 y Fm(\025;r)946 3396 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)1379 3411 y Fm(z)1418 3396 y Fr(\))1493 3370 y(~)1473 3396 y Fo( )1536 3411 y Fm(\025;r)1631 3392 y Fc(0)1657 3396 y Fr(\()p Fo(t)p Fr(\))1768 3286 y Fk(E)1863 3396 y Fo(d\025)470 3669 y Fr(=)83 b Fo(\031)704 3533 y Fk(Z)760 3759 y Fe(R)p 828 3542 655 4 v 828 3669 a Fo(\036)886 3684 y Fm(r)941 3558 y Fk(\020)1008 3669 y Fl(~)-64 b Ff(\034)1066 3626 y Fx(\()p Fm(\017)p Fx(\))1153 3669 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))1423 3558 y Fk(\021)1499 3669 y Fo(\036)1557 3684 y Fm(r)1591 3665 y Fc(0)1634 3558 y Fk(\020)1701 3669 y Fl(~)h Ff(\034)1759 3626 y Fx(\()p Fm(\017)p Fx(\))1846 3669 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))2116 3558 y Fk(\021)2193 3498 y( )2282 3601 y Fo(m)2367 3565 y Fx(2)2428 3601 y Fr(+)22 b Fn(j)p Fl(p)2616 3616 y Fy(?)2675 3601 y Fn(j)2703 3553 y Fx(2)p 2282 3646 462 4 v 2395 3737 a Fo(\024)p Fr(\()p Fl(p)p Fr(\))2589 3708 y Fx(2)2775 3669 y Fr(+)g(1)2922 3498 y Fk(!)629 3941 y Fn(\002)723 3830 y Fk(D)804 3915 y Fr(~)784 3941 y Fo( )847 3956 y Fm(\024)p Fx(\()p Fi(p)p Fx(\))p Fm(;r)1046 3941 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()o Fl(1)23 b Fn(\000)f Fo(\013)1478 3956 y Fm(z)1518 3941 y Fr(\))1592 3915 y(~)1573 3941 y Fo( )1636 3956 y Fm(\024)p Fx(\()p Fi(p)p Fx(\))p Fm(;r)1830 3937 y Fc(0)1857 3941 y Fr(\()p Fo(t)p Fr(\))1968 3830 y Fk(E)2045 3941 y Fo(dp)2145 3956 y Fm(z)2185 3941 y Fo(;)862 b Fr(\(A61\))83 4319 y(where)34 b(w)m(e)f(ha)m(v)m(e)h(substituted)g(bac) m(k)f(from)f Fo(\025)g Fr(to)g Fo(p)1959 4334 y Fm(z)1999 4319 y Fr(.)44 b(By)33 b(virtue)f(of)g(the)h(iden)m(tities)1082 4675 y Fl(1)23 b Fn(\000)f Fo(\013)1322 4690 y Fm(z)1390 4675 y Fr(=)27 b Fo(\015)1549 4634 y Fx(0)1605 4595 y Fk(\000)1651 4675 y Fo(\015)1707 4634 y Fx(0)1769 4675 y Fn(\000)22 b Fo(\015)1924 4634 y Fx(3)1964 4595 y Fk(\001)2037 4675 y Fr(=)28 b Fo(\015)2197 4634 y Fx(0)2241 4678 y Fo(=)2236 4675 y(n;)753 b Fr(\(A62\))83 5032 y(whic)m(h)33 b(is)f(a)h(hermitean)e(matrix,)g(and)1440 5391 y Fo(=)1435 5388 y(n)1493 5343 y Fx(2)1560 5388 y Fr(=)d Fo(n)1722 5347 y Fx(2)1789 5388 y Fr(=)g(0)p Fo(;)1105 b Fr(\(A63\))1653 5637 y(29)p eop %%Page: 30 30 30 29 bop 432 573 a Fr(w)m(e)33 b(\014nd)825 682 y Fk(D)906 766 y Fr(~)886 793 y Fo( )949 808 y Fm(\025;r)1048 793 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)1481 808 y Fm(z)1520 793 y Fr(\))1595 766 y(~)1575 793 y Fo( )1638 808 y Fm(\025;r)1733 789 y Fc(0)1759 793 y Fr(\()p Fo(t)p Fr(\))1870 682 y Fk(E)908 1065 y Fr(=)1067 895 y Fk(* )1226 1065 y Fl(1)f Fr(+)1412 998 y Fo(e)1462 1001 y(=)1457 998 y(n)1528 957 y Fr(~)1535 989 y Fo(=)1516 998 y(A)p Fr(\(0\))p 1412 1042 302 4 v 1490 1133 a(2)p Fo(\017\025)1723 895 y Fk(!)1825 1065 y Fr(~)-55 b Fo(u)1875 1080 y Fm(r)1912 1065 y Fr(\()p Fo(\025)p Fr(\))p Fo(;)17 b(\015)2145 1024 y Fx(0)2189 1068 y Fo(=)2184 1065 y(n)2259 895 y Fk( )2338 1065 y Fl(1)22 b Fr(+)2524 998 y Fo(e)2574 1001 y(=)2569 998 y(n)2640 957 y Fr(~)2647 989 y Fo(=)2628 998 y(A)p Fr(\(0\))p 2524 1042 V 2602 1133 a(2)p Fo(\017\025)2835 895 y Fk(!)2937 1065 y Fr(~)-55 b Fo(u)2987 1080 y Fm(r)3021 1061 y Fc(0)3047 1065 y Fr(\()p Fo(\025)p Fr(\))3180 895 y Fk(+)908 1397 y Fr(=)1067 1227 y Fk(* )1226 1397 y Fl(1)22 b Fr(+)1412 1330 y Fo(e)1462 1333 y(=)1457 1330 y(n)1528 1289 y Fr(~)1535 1321 y Fo(=)1516 1330 y(A)p Fr(\(0\))p 1412 1374 V 1490 1466 a(2)p Fo(\017\025)1723 1227 y Fk(!)1825 1397 y Fr(~)-55 b Fo(u)1875 1412 y Fm(r)1912 1397 y Fr(\()p Fo(\025)p Fr(\))p Fo(;)17 b(\015)2145 1356 y Fx(0)2189 1400 y Fo(=)2184 1397 y(n)7 b Fr(~)-55 b Fo(u)2299 1412 y Fm(r)2333 1393 y Fc(0)2358 1397 y Fr(\()p Fo(\025)p Fr(\))2491 1227 y Fk(+)908 1730 y Fr(=)1067 1559 y Fk(*)1147 1730 y Fo(\015)1203 1688 y Fx(0)1247 1733 y Fo(=)1243 1730 y(n)1317 1559 y Fk( )1396 1730 y Fl(1)23 b Fr(+)1583 1662 y Fo(e)1633 1665 y(=)1628 1662 y(n)1698 1621 y Fr(~)1705 1653 y Fo(=)1686 1662 y(A)p Fr(\(0\))p 1583 1707 V 1661 1798 a(2)p Fo(\017\025)1894 1559 y Fk(!)1995 1730 y Fr(~)-55 b Fo(u)2045 1745 y Fm(r)2083 1730 y Fr(\()p Fo(\025)p Fr(\))p Fo(;)22 b Fr(~)-54 b Fo(u)2316 1745 y Fm(r)2350 1726 y Fc(0)2375 1730 y Fr(\()p Fo(\025)p Fr(\))2508 1559 y Fk(+)908 1972 y Fr(=)1067 1891 y Fk(\012)1114 1972 y Fo(\015)1170 1931 y Fx(0)1214 1975 y Fo(=)1209 1972 y(n)7 b Fr(~)-55 b Fo(u)1324 1987 y Fm(r)1361 1972 y Fr(\()p Fo(\025)p Fr(\))p Fo(;)23 b Fr(~)-55 b Fo(u)1594 1987 y Fm(r)1628 1968 y Fc(0)1653 1972 y Fr(\()p Fo(\025)p Fr(\))1786 1891 y Fk(\013)908 2117 y Fr(=)1067 2037 y Fk(\012)1120 2117 y Fr(~)g Fo(u)1170 2132 y Fm(r)1207 2117 y Fr(\()p Fo(\025)p Fr(\))p Fo(;)17 b(\015)1440 2076 y Fx(0)1484 2120 y Fo(=)1479 2117 y(n)7 b Fr(~)-55 b Fo(u)1594 2132 y Fm(r)1628 2113 y Fc(0)1653 2117 y Fr(\()p Fo(\025)p Fr(\))1786 2037 y Fk(\013)908 2300 y Fr(=)1067 2189 y Fk(D)1128 2300 y Fo(u)1184 2315 y Fm(r)1221 2300 y Fr(\()8 b Fl(~)-64 b Ff(\034)1325 2257 y Fx(\()p Fm(\017)p Fx(\))1412 2300 y Fr(\()p Fo(\025)p Fr(\)\))p Fo(;)17 b(\015)1683 2259 y Fx(0)1727 2303 y Fo(=)1722 2300 y(nu)1836 2315 y Fm(r)1870 2296 y Fc(0)1896 2300 y Fr(\()8 b Fl(~)-64 b Ff(\034)1999 2257 y Fx(\()p Fm(\017)p Fx(\))2087 2300 y Fr(\()p Fo(\025)p Fr(\)\))2258 2189 y Fk(E)2335 2300 y Fo(:)1061 b Fr(\(A64\))432 2565 y(In)34 b(order)h(to)f(ev)-5 b(aluate)34 b(this)g(scalar)g(pro)s(duct,) h(\014x)g Fo(\025)f Fr(and)h(set)g Fl(p)c Fr(:=)38 b Fl(~)-64 b Ff(\034)3052 2522 y Fx(\()p Fm(\017)p Fx(\))3139 2565 y Fr(\()p Fo(\025)p Fr(\).)49 b(W)-8 b(e)35 b(will)432 2685 y(mak)m(e)d(use)i(of)e(the)h(4)22 b Fn(\002)h Fr(4)32 b(matrices)1745 2649 y Fx(15)645 2948 y Fr(\003)713 2963 y Fy(\006)771 2948 y Fr(\()p Fl(p)p Fr(\))c(:=)1078 2871 y Fn(\006)1155 2884 y Fo(=)1155 2871 y(p)22 b Fr(+)g Fo(m)p 1078 2925 332 4 v 1177 3016 a Fr(2)p Fo(m)1447 2948 y Fn(\021)1562 2880 y(\006)17 b Fr(\()p Fo(\015)1750 2844 y Fx(0)1790 2880 y Fo(E)6 b Fr(\()p Fl(p)p Fr(\))22 b Fn(\000)h Fo(\015)2184 2844 y Fx(1)2223 2880 y Fo(p)2272 2895 y Fm(x)2338 2880 y Fn(\000)g Fo(\015)2494 2844 y Fx(2)2533 2880 y Fo(p)2582 2895 y Fm(y)2646 2880 y Fn(\000)g Fo(\015)2802 2844 y Fx(3)2841 2880 y Fo(p)2890 2895 y Fm(z)2930 2880 y Fr(\))f(+)g Fo(m)p 1562 2925 1611 4 v 2301 3016 a Fr(2)p Fo(m)3183 2948 y(;)213 b Fr(\(A65\))432 3193 y(whic)m(h)33 b(ob)s(ey)1603 3313 y(\003)1671 3328 y Fm(\017)1700 3336 y Fh(r)1738 3313 y Fr(\()p Fl(p)p Fr(\))p Fo(u)1932 3328 y Fm(r)1970 3313 y Fr(\()p Fl(p)p Fr(\))28 b(=)f Fo(u)2295 3328 y Fm(r)2333 3313 y Fr(\()p Fl(p)p Fr(\))p Fo(;)925 b Fr(\(A66\))432 3488 y(and)32 b(the)h(fact)g(that)f(for)g(an)m(y)h(four-v)m(ector)g Fo(a)p Fr(,)1808 3710 y Fo(=)1807 3707 y(a)1858 3662 y Fy(y)1921 3707 y Fr(=)28 b Fo(\015)2081 3666 y Fx(0)2122 3710 y Fo(=)2120 3707 y(a)q(\015)2228 3666 y Fx(0)2267 3707 y Fo(:)1129 b Fr(\(A67\))432 3927 y(This)32 b(giv)m(es)741 4058 y Fk(D)822 4143 y Fr(~)802 4169 y Fo( )865 4184 y Fm(\025;r)964 4169 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()p Fl(1)22 b Fn(\000)h Fo(\013)1397 4184 y Fm(z)1436 4169 y Fr(\))1511 4143 y(~)1491 4169 y Fo( )1554 4184 y Fm(\025;r)1649 4165 y Fc(0)1676 4169 y Fr(\()p Fo(t)p Fr(\))1787 4058 y Fk(E)1931 4169 y Fr(=)2089 4088 y Fk(\012)2136 4169 y Fr(\003)2204 4184 y Fm(\017)2237 4169 y Fr(\()p Fl(p)p Fr(\))p Fo(u)2431 4184 y Fm(r)2468 4169 y Fr(\()p Fl(p)p Fr(\))p Fo(;)17 b(\015)2706 4128 y Fx(0)2751 4172 y Fo(=)2746 4169 y(n)p Fr(\003)2872 4184 y Fm(\017)2904 4169 y Fr(\()p Fl(p)p Fr(\))p Fo(u)3098 4184 y Fm(r)3132 4165 y Fc(0)3158 4169 y Fr(\()p Fl(p)p Fr(\))3296 4088 y Fk(\013)1931 4352 y Fr(=)2089 4271 y Fk(\012)2136 4352 y Fo(u)2192 4367 y Fm(r)2230 4352 y Fr(\()p Fl(p)p Fr(\))p Fo(;)g(\015)2468 4310 y Fx(0)2507 4352 y Fr(\003)2575 4367 y Fm(\017)2608 4352 y Fr(\()p Fl(p)p Fr(\))2751 4355 y Fo(=)2746 4352 y(n)p Fr(\003)2872 4367 y Fm(\017)2904 4352 y Fr(\()p Fl(p)p Fr(\))p Fo(u)3098 4367 y Fm(r)3132 4348 y Fc(0)3158 4352 y Fr(\()p Fl(p)p Fr(\))3296 4271 y Fk(\013)1931 4497 y Fr(=)2089 4416 y Fk(\012)2136 4497 y Fo(u)2192 4512 y Fm(r)2230 4497 y Fr(\()p Fl(p)p Fr(\))p Fo(;)g(\015)2468 4456 y Fx(0)2507 4497 y Fo(M)10 b(u)2667 4512 y Fm(r)2701 4493 y Fc(0)2728 4497 y Fr(\()p Fl(p)p Fr(\))2866 4416 y Fk(\013)3423 4497 y Fr(\(A68\))432 4717 y(with)32 b(the)h(4)22 b Fn(\002)g Fr(4)33 b(matrix)1122 4936 y Fo(M)94 b Fr(:=)83 b(\003)1564 4951 y Fm(\017)1596 4936 y Fr(\()p Fl(p)p Fr(\))1739 4939 y Fo(=)1734 4936 y(n)q Fr(\003)1861 4951 y Fm(\017)1893 4936 y Fr(\()p Fl(p)p Fr(\))1323 5136 y(=)1506 5060 y Fo(\017)1545 5073 y(=)1545 5060 y(p)22 b Fr(+)g Fo(m)p 1506 5114 294 4 v 1586 5205 a Fr(2)p Fo(m)1831 5139 y(=)1826 5136 y(n)1911 5060 y(\017)1950 5073 y(=)1950 5060 y(p)g Fr(+)g Fo(m)p 1911 5114 V 1991 5205 a Fr(2)p Fo(m)1323 5370 y Fr(=)1568 5302 y(1)p 1506 5347 174 4 v 1506 5438 a(4)p Fo(m)1640 5409 y Fx(2)1706 5289 y Fk(\002)1748 5383 y Fo(=)1747 5370 y(p)1801 5373 y(=)1796 5370 y(n)1855 5383 y(=)1855 5370 y(p)g Fr(+)g Fo(\017m)2165 5289 y Fk(\000)2211 5383 y Fo(=)2211 5370 y(p)2264 5373 y(=)2260 5370 y(n)g Fr(+)2443 5373 y Fo(=)2438 5370 y(n)2496 5383 y(=)2496 5370 y(p)2545 5289 y Fk(\001)2613 5370 y Fr(+)g Fo(m)2796 5328 y Fx(2)2841 5373 y Fo(=)2836 5370 y(n)2894 5289 y Fk(\003)2952 5370 y Fo(:)444 b Fr(\(A69\))2002 5637 y(30)p eop %%Page: 31 31 31 30 bop 83 573 a Fr(Using)1327 696 y Fo(=)1326 693 y(a)1377 683 y(=)1381 693 y(b)26 b Fr(+)1546 683 y Fo(=)1550 693 y(b)1596 696 y(=)1595 693 y(a)i Fr(=)f(2\()p Fo(ab)p Fr(\))p Fl(1)q Fo(;)996 b Fr(\(A70\))83 862 y(whic)m(h)33 b(is)f(v)-5 b(alid)31 b(for)h(an)m(y)h(four-v)m(ectors)h Fo(a)e Fr(and)h Fo(b)g Fr(of)f(real)g(n)m(um)m(b)s(ers,)h(and)1406 1081 y Fo(=)1406 1068 y(p)1455 1027 y Fx(2)1522 1068 y Fr(=)28 b Fo(p)1675 1027 y Fx(2)1742 1068 y Fr(=)g Fo(m)1931 1027 y Fx(2)1970 1068 y Fo(;)1077 b Fr(\(A71\))83 1274 y(w)m(e)34 b(con)m(tin)m(ue:)123 1510 y Fo(M)39 b Fr(=)432 1443 y(1)p 369 1488 174 4 v 369 1579 a(4)p Fo(m)503 1550 y Fx(2)569 1430 y Fk(\002)611 1510 y Fr(2\()p Fo(np)p Fr(\))843 1523 y Fo(=)843 1510 y(p)22 b Fn(\000)h Fo(m)1099 1469 y Fx(2)1143 1513 y Fo(=)1138 1510 y(n)g Fr(+)f(2)p Fo(\017m)p Fr(\()p Fo(np)p Fr(\))g(+)g Fo(m)1878 1469 y Fx(2)1923 1513 y Fo(=)1918 1510 y(n)1976 1430 y Fk(\003)2046 1510 y Fr(=)2159 1443 y Fo(\017)p Fr(\()p Fo(np)p Fr(\))p 2159 1488 223 4 v 2228 1579 a Fo(m)2392 1510 y Fr(\003)2460 1525 y Fm(\017)2492 1510 y Fr(\()p Fl(p)p Fr(\))28 b(=)2771 1443 y Fo(\017\024)p Fr(\()p Fl(p)p Fr(\))p 2771 1488 234 4 v 2845 1579 a Fo(m)3015 1510 y Fr(\003)3083 1525 y Fm(\017)3115 1510 y Fr(\()p Fl(p)p Fr(\))p Fo(:)3074 1662 y Fr(\(A72\))83 1782 y(Hence,)183 1952 y Fk(D)263 2037 y Fr(~)244 2063 y Fo( )307 2078 y Fm(\025;r)406 2063 y Fr(\()p Fo(t)p Fr(\))p Fo(;)17 b Fr(\()o Fl(1)23 b Fn(\000)f Fo(\013)838 2078 y Fm(z)878 2063 y Fr(\))953 2037 y(~)933 2063 y Fo( )996 2078 y Fm(\025;r)1091 2059 y Fc(0)1117 2063 y Fr(\()p Fo(t)p Fr(\))1228 1952 y Fk(E)1317 2063 y Fr(=)1430 1995 y Fo(\017\024)p Fr(\()p Fl(p)p Fr(\))p 1430 2040 V 1504 2131 a Fo(m)1691 1982 y Fk(\012)1738 2063 y Fo(u)1794 2078 y Fm(r)1831 2063 y Fr(\()p Fl(p)p Fr(\))p Fo(;)17 b(\015)2069 2022 y Fx(0)2108 2063 y Fr(\003)2176 2078 y Fm(\017)2209 2063 y Fr(\()p Fl(p)p Fr(\))p Fo(u)2403 2078 y Fm(r)2437 2059 y Fc(0)2463 2063 y Fr(\()p Fl(p)p Fr(\))2601 1982 y Fk(\013)2676 2063 y Fr(=)1465 2239 y Fo(\017\024)p Fr(\()p Fl(p)p Fr(\))p 1465 2284 V 1539 2375 a Fo(m)1725 2226 y Fk(\012)1772 2306 y Fo(u)1828 2321 y Fm(r)1866 2306 y Fr(\()p Fl(p)p Fr(\))p Fo(;)g(\015)2104 2265 y Fx(0)2143 2306 y Fo(u)2199 2321 y Fm(r)2233 2302 y Fc(0)2259 2306 y Fr(\()p Fl(p)p Fr(\))2397 2226 y Fk(\013)2472 2306 y Fr(=)27 b Fo(\016)2618 2321 y Fm(r)n(;r)2702 2302 y Fc(0)2749 2239 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p 2738 2284 217 4 v 2738 2375 a Fo(E)6 b Fr(\()p Fl(p)p Fr(\))2964 2306 y Fo(;)83 b Fr(\(A73\))83 2563 y(where)669 2603 y Fk(\012)716 2683 y Fo(u)772 2698 y Fm(r)809 2683 y Fr(\()p Fl(p)p Fr(\))p Fo(;)17 b(\015)1047 2642 y Fx(0)1087 2683 y Fo(u)1143 2698 y Fm(r)1177 2679 y Fc(0)1203 2683 y Fr(\()p Fl(p)p Fr(\))1341 2603 y Fk(\013)1415 2683 y Fn(\021)29 b Fo(u)1577 2698 y Fm(r)1614 2683 y Fr(\()p Fl(p)p Fr(\))1752 2642 y Fy(y)1788 2683 y Fo(\015)1844 2642 y Fx(0)1883 2683 y Fo(u)1939 2698 y Fm(r)1973 2679 y Fc(0)1999 2683 y Fr(\()p Fl(p)p Fr(\))f(=)f Fo(\017)2307 2698 y Fm(r)2346 2683 y Fo(\016)2389 2698 y Fm(r)n(;r)2473 2679 y Fc(0)2574 2616 y Fo(m)p 2509 2661 V 2509 2752 a(E)6 b Fr(\()p Fl(p)p Fr(\))3074 2683 y(\(A74\))83 2912 y(w)m(as)34 b(exploited.)686 2876 y Fx(15)803 2912 y Fr(Therefore,)g(w)m(e)g(obtain)d(from)g(\(A61\):)620 3035 y Fk(D)694 3119 y Fr(~)681 3146 y Fo(\036)739 3161 y Fm(r)777 3146 y Fr(\()p Fo(\017)854 3161 y Fm(r)892 3146 y Fl(p)954 3161 y Fy(?)1013 3146 y Fo(;)17 b Fn(\001)p Fr(\))p Fo(;)1179 3119 y Fr(~)1167 3146 y Fo(\036)1225 3161 y Fm(r)1259 3142 y Fc(0)1284 3146 y Fr(\()p Fo(\017)1361 3161 y Fm(r)1395 3142 y Fc(0)1422 3146 y Fl(p)1484 3161 y Fy(?)1543 3146 y Fo(;)g Fn(\001)p Fr(\))1653 3035 y Fk(E)1714 3215 y Fm(L)1762 3196 y Fg(2)703 3383 y Fr(=)83 b Fo(\016)905 3398 y Fm(r)n(;r)989 3379 y Fc(0)1032 3383 y Fo(\031)1107 3248 y Fk(Z)1163 3473 y Fe(R)p 1231 3257 655 4 v 1231 3383 a Fo(\036)1289 3398 y Fm(r)1344 3273 y Fk(\020)1411 3383 y Fl(~)-64 b Ff(\034)1468 3340 y Fx(\()p Fm(\017)p Fx(\))1556 3383 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))1826 3273 y Fk(\021)1902 3383 y Fo(\036)1960 3398 y Fm(r)1994 3379 y Fc(0)2037 3273 y Fk(\020)2104 3383 y Fl(~)g Ff(\034)2162 3340 y Fx(\()p Fm(\017)p Fx(\))2249 3383 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))2519 3273 y Fk(\021)862 3681 y Fn(\002)956 3511 y Fk( )1045 3613 y Fo(m)1130 3577 y Fx(2)1192 3613 y Fr(+)22 b Fn(j)p Fl(p)1380 3628 y Fy(?)1439 3613 y Fn(j)1467 3565 y Fx(2)p 1045 3658 462 4 v 1159 3749 a Fo(\024)p Fr(\()p Fl(p)p Fr(\))1353 3720 y Fx(2)1538 3681 y Fr(+)g(1)1685 3511 y Fk(!)1801 3613 y Fo(\024)p Fr(\()p Fl(p)p Fr(\))p 1790 3658 217 4 v 1790 3749 a Fo(E)6 b Fr(\()p Fl(p)p Fr(\))2033 3681 y Fo(dp)2133 3696 y Fm(z)703 3974 y Fr(=)83 b(2)p Fo(\031)t(\016)1013 3989 y Fm(r)n(;r)1097 3970 y Fc(0)1139 3839 y Fk(Z)1195 4064 y Fe(R)p 1263 3848 655 4 v 1263 3974 a Fo(\036)1321 3989 y Fm(r)1376 3864 y Fk(\020)1443 3974 y Fl(~)-64 b Ff(\034)1501 3931 y Fx(\()p Fm(\017)p Fx(\))1588 3974 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))1858 3864 y Fk(\021)1934 3974 y Fo(\036)1992 3989 y Fm(r)2026 3970 y Fc(0)2069 3864 y Fk(\020)2136 3974 y Fl(~)h Ff(\034)2194 3931 y Fx(\()p Fm(\017)p Fx(\))2281 3974 y Fr(\()p Fo(\024)p Fr(\()p Fl(p)p Fr(\)\))2551 3864 y Fk(\021)2644 3974 y Fo(dp)2744 3989 y Fm(z)703 4233 y Fr(=)83 b(2)p Fo(\031)t(\016)1013 4248 y Fm(r)n(;r)1097 4229 y Fc(0)1139 4097 y Fk(Z)1195 4323 y Fe(R)1263 4233 y Fn(j)p Fo(\036)1349 4248 y Fm(r)1387 4233 y Fr(\()p Fo(\017)1464 4248 y Fm(r)1502 4233 y Fl(p)1564 4248 y Fy(?)1623 4233 y Fo(;)17 b(p)1716 4248 y Fm(z)1756 4233 y Fr(\))p Fn(j)1821 4185 y Fx(2)1894 4233 y Fo(dp)1994 4248 y Fm(z)2034 4233 y Fo(;)1013 b Fr(\(A75\))83 4492 y(whic)m(h)29 b(con\014rms)h(\(A10\))e(also)g(in)g(the)i(case)f Fo(\017)1729 4507 y Fm(r)1795 4492 y Fr(=)f Fo(\017)1938 4507 y Fm(r)1972 4488 y Fc(0)1999 4492 y Fr(.)42 b(This)29 b(completes)g(the)g(pro)s(of.)p 3250 4492 4 66 v 3254 4429 59 4 v 3254 4492 V 3312 4492 4 66 v 229 4683 a(W)-8 b(e)33 b(are)g(left)f(with)g(sho)m(wing:)83 4875 y Fl(Lemma)37 b(9.)42 b Fp(L)-5 b(et)35 b Fo(f)j Fn(2)29 b(C)1010 4890 y Fx(0)1066 4875 y Fr(\([0)p Fo(;)17 b Fr(1])22 b Fn(\002)g Fj(R)1487 4838 y Fx(2)1533 4875 y Fo(;)17 b Fj(C)i Fr(\))6 b Fp(.)45 b(Then)228 5129 y Fo(I)271 5144 y Fm(\016)337 5129 y Fr(:=)467 4994 y Fk(Z)523 5219 y Fe(R)571 5200 y Fg(2)626 5129 y Fo(\021)674 5144 y Fm(\016)712 5129 y Fr(\()p Fo(x)22 b Fn(\000)h Fo(x)982 5088 y Fy(0)1006 5129 y Fr(\))p Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)1380 5088 y Fy(0)1403 5129 y Fr(\))g Fo(dx)g(dx)1687 5088 y Fy(0)1738 5129 y Fn(!)1865 4994 y Fk(Z)1920 5219 y Fe(R)1989 5129 y Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\))g Fo(dx)27 b(;)45 b(\016)31 b Fn(!)c Fr(0)2843 5088 y Fx(+)2902 5129 y Fo(;)145 b Fr(\(A76\))83 5388 y Fp(with)35 b Fo(\021)343 5403 y Fm(\016)416 5388 y Fp(fr)-5 b(om)41 b Fr(\(A55\))p Fp(.)1653 5637 y Fr(31)p eop %%Page: 32 32 32 31 bop 432 573 a Fp(Pr)-5 b(o)g(of.)48 b Fr(W)-8 b(e)33 b(\014rst)g(collect)f(some)g(prop)s(erties)h(of)f(the)h(functions)f Fo(\021)2860 588 y Fm(\016)2898 573 y Fr(:)1465 798 y Fn(8)p Fo(\016)g(>)c Fr(0)17 b Fo(;)33 b(x)28 b Fn(2)g Fj(R)66 b Fr(:)56 b Fo(\021)2260 813 y Fm(\016)2298 798 y Fr(\()p Fo(x)p Fr(\))27 b Fo(>)h Fr(0)p Fo(;)787 b Fr(\(A77\))1505 1076 y Fn(8)p Fo(\016)32 b(>)c Fr(0)55 b(:)1925 940 y Fk(Z)1980 1166 y Fe(R)2049 1076 y Fo(\021)2097 1091 y Fm(\016)2135 1076 y Fr(\()p Fo(x)p Fr(\))17 b Fo(dx)28 b Fr(=)f(1)p Fo(;)827 b Fr(\(A78\))432 1303 y(and)1169 1452 y Fn(8)p Fo(\015)33 b(>)28 b Fr(0)55 b(:)1599 1316 y Fk(Z)1654 1542 y Fe(R)p Fy(n)p Fx([)p Fy(\000)p Fm(\015)t(;\015)t Fx(])1952 1452 y Fo(\021)2000 1467 y Fm(\016)2038 1452 y Fr(\()p Fo(x)p Fr(\))17 b Fo(dx)28 b Fn(!)f Fr(0)h Fo(;)44 b(\016)32 b Fn(!)27 b Fr(0)2846 1410 y Fx(+)2905 1452 y Fo(:)491 b Fr(\(A79\))432 1694 y(Set)1295 1843 y Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))28 b(:=)1720 1707 y Fk(Z)1776 1933 y Fe(R)1844 1843 y Fo(\021)1892 1858 y Fm(\016)1930 1843 y Fr(\()p Fo(x)23 b Fn(\000)g Fo(x)2201 1802 y Fy(0)2224 1843 y Fr(\))p Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)2598 1802 y Fy(0)2622 1843 y Fr(\))g Fo(dx)2783 1802 y Fy(0)3423 1843 y Fr(\(A80\))432 2070 y(for)32 b Fo(\016)f(>)d Fr(0,)k(so)h(that) 1669 2238 y Fo(I)1712 2253 y Fm(\016)1777 2238 y Fr(=)1881 2102 y Fk(Z)1936 2328 y Fe(R)2005 2238 y Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))27 b Fo(dx:)991 b Fr(\(A81\))432 2471 y(The)34 b(compactness)h(of)f(supp)17 b Fo(f)45 b Fr(implies)31 b(the)j(existence)h(of)f(a)f(compact)h(in)m(terv)-5 b(al)32 b([)p Fo(a;)17 b(b)p Fr(])432 2591 y(suc)m(h)34 b(that)1322 2716 y Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))28 b(=)f(0)97 b(for)g Fo(\016)32 b(>)c Fr(0)17 b Fo(;)32 b(x)c Fn(62)h Fr([)p Fo(a;)17 b(b)p Fr(])p Fo(:)644 b Fr(\(A82\))432 2896 y(Therefore,)1655 3084 y Fo(I)1698 3099 y Fm(\016)1764 3084 y Fr(=)1868 2949 y Fk(Z)1967 2975 y Fm(b)1923 3174 y(a)2018 3084 y Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))28 b Fo(dx:)977 b Fr(\(A83\))432 3311 y(Note)32 b(that)795 3560 y Fn(j)p Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))p Fn(j)27 b(\024)1250 3424 y Fk(Z)1306 3650 y Fe(R)1374 3560 y Fo(\021)1422 3575 y Fm(\016)1460 3560 y Fr(\()p Fo(x)c Fn(\000)g Fo(x)1731 3519 y Fy(0)1754 3560 y Fr(\))17 b Fn(j)o Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)2172 3519 y Fy(0)2196 3560 y Fr(\))p Fn(j)33 b Fo(dx)2401 3519 y Fy(0)2452 3560 y Fn(\024)28 b(jj)p Fo(f)11 b Fn(j)o(j)2727 3589 y Fy(1)2829 3560 y Fo(<)28 b Fn(1)p Fo(:)363 b Fr(\(A84\))432 3832 y(W)-8 b(e)33 b(claim)d(that)1263 3957 y Fn(8)p Fo(x)f Fn(2)f Fr([)p Fo(a;)17 b(b)p Fr(])56 b(:)83 b(lim)1824 4024 y Fm(\016)r Fy(!)p Fx(0)1963 4005 y Fg(+)2031 3957 y Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))28 b(=)f Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\))p Fo(:)585 b Fr(\(A85\))432 4171 y(The)33 b(statemen)m(t)g(in)f(the)h(lemma)d (follo)m(ws)i(then)h(b)m(y)g(dominated)f(con)m(v)m(ergence.)578 4293 y(In)h(order)g(to)f(pro)m(v)m(e)i(\(A85\))o(,)f(\014x)g Fo(x)28 b Fn(2)g Fr([)p Fo(a;)17 b(b)p Fr(].)44 b(F)-8 b(or)32 b(an)m(y)h Fo(\015)5 b(;)17 b(\016)32 b(>)27 b Fr(0,)33 b(w)m(e)g(estimate)636 4575 y Fn(j)p Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))22 b Fn(\000)h Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\))p Fn(j)83 b Fr(=)1705 4431 y Fk(\014)1705 4491 y(\014)1705 4550 y(\014)1705 4610 y(\014)1739 4440 y(Z)1794 4665 y Fe(R)1863 4575 y Fo(\021)1911 4590 y Fm(\016)1949 4575 y Fr(\()p Fo(x)22 b Fn(\000)h Fo(x)2219 4534 y Fy(0)2243 4575 y Fr(\))17 b([)o Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)2660 4534 y Fy(0)2684 4575 y Fr(\))22 b Fn(\000)h Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\)])33 b Fo(dx)3392 4534 y Fy(0)3415 4431 y Fk(\014)3415 4491 y(\014)3415 4550 y(\014)3415 4610 y(\014)1546 4848 y Fr(=)1705 4703 y Fk(\014)1705 4763 y(\014)1705 4823 y(\014)1705 4883 y(\014)1739 4712 y(Z)1794 4938 y Fe(R)1863 4848 y Fo(\021)1911 4863 y Fm(\016)1949 4848 y Fr(\()p Fo(y)t Fr(\))17 b([)n Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)23 b Fn(\000)g Fo(y)t Fr(\))e Fn(\000)i Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\)])32 b Fo(dy)3333 4703 y Fk(\014)3333 4763 y(\014)3333 4823 y(\014)3333 4883 y(\014)1545 5111 y Fn(\024)83 b Fr(2)17 b Fn(jj)o Fo(f)11 b Fn(jj)1940 5140 y Fy(1)2032 4976 y Fk(Z)2087 5201 y Fe(R)p Fy(n)p Fx([)p Fy(\000)p Fm(\015)t(;\015)t Fx(])2385 5111 y Fo(\021)2433 5126 y Fm(\016)2471 5111 y Fr(\()p Fo(y)t Fr(\))17 b Fo(dy)1705 5326 y Fr(+)81 b(sup)1798 5411 y Fm(r)r Fy(2)p Fx([)p Fy(\000)p Fm(\015)t(;\015)t Fx(])2090 5326 y Fn(j)o Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)23 b Fr(+)f Fo(r)s Fr(\))g Fn(\000)g Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\))p Fn(j)f Fo(;)190 b Fr(\(A86\))2002 5637 y(32)p eop %%Page: 33 33 33 32 bop 83 573 a Fr(where)35 b(\(A78\))f(w)m(as)h(emplo)m(y)m(ed)f(t) m(wice.)49 b(Cho)s(ose)35 b(an)m(y)g Fo(\017)30 b(>)g Fr(0.)48 b(Since)35 b Fo(f)44 b Fr(is)34 b(con)m(tin)m(uous,)83 693 y(w)m(e)g(\014nd)f Fo(\015)5 b(;)17 b(\013)28 b(>)f Fr(0)33 b(suc)m(h)h(that)478 936 y(sup)414 1022 y Fm(r)r Fy(2)p Fx([)p Fy(\000)p Fm(\015)t(;\015)t Fx(])705 936 y Fn(j)p Fo(f)11 b Fr(\()p Fo(\016)n(;)17 b(x;)g(x)23 b Fr(+)f Fo(r)s Fr(\))f Fn(\000)i Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\))p Fn(j)27 b Fo(<)1951 868 y(\017)p 1946 913 49 4 v 1946 1004 a Fr(2)2103 936 y(for)97 b Fo(\016)31 b Fn(2)e Fr(\(0)p Fo(;)17 b(\013)q Fr(\))p Fo(:)329 b Fr(\(A87\))83 1236 y(In)33 b(view)g(of)39 b(\(A79\))o(,)33 b(w)m(e)h(\014nd)f Fo(\014)g Fn(2)28 b Fr(\(0)p Fo(;)17 b(\013)q Fr(\))32 b(with)728 1509 y(2)17 b Fn(j)o(j)p Fo(f)11 b Fn(j)o(j)963 1538 y Fy(1)1054 1374 y Fk(Z)1110 1599 y Fe(R)p Fy(n)p Fx([)p Fy(\000)p Fm(\015)t(;\015)t Fx(])1408 1509 y Fo(\021)1456 1524 y Fm(\016)1494 1509 y Fr(\()p Fo(y)t Fr(\))17 b Fo(dy)29 b(<)1885 1442 y(\017)p 1881 1486 V 1881 1578 a Fr(2)2037 1509 y(for)97 b Fo(\016)32 b Fn(2)c Fr(\(0)p Fo(;)17 b(\014)6 b Fr(\))p Fo(:)397 b Fr(\(A88\))83 1798 y(Therefore,)836 1923 y Fn(j)o Fo(g)t Fr(\()p Fo(\016)n(;)17 b(x)p Fr(\))22 b Fn(\000)h Fo(f)11 b Fr(\(0)p Fo(;)17 b(x;)g(x)p Fr(\))p Fn(j)27 b Fo(<)g(\017)98 b Fr(for)f Fo(\016)32 b Fn(2)c Fr(\(0)p Fo(;)17 b(\014)6 b Fr(\))p Fo(;)505 b Fr(\(A89\))83 2103 y(and)33 b(the)g(lemma)d(is)i(pro)m(v)m(ed.)p 3250 2103 4 66 v 3254 2041 59 4 v 3254 2103 V 3312 2103 4 66 v 83 2447 a Fq(References)83 2633 y Fx(1)139 2669 y Fr(D.)g(M.)h(W)-8 b(olk)m(o)m(w,)33 b(Z.)f(Ph)m(ys.)i Fl(94)p Fr(,)f(250)f(\(1935\).)83 2845 y Fx(2)139 2881 y Fr(T.)h(S.)g(Bro)m(wn)g(and)g(T.)g(W.)f(B.)h(Kibble,)f(Ph)m(ys.)i (Rev.)f Fl(133)p Fr(,)g(A705)f(\(1964\).)83 3056 y Fx(3)139 3092 y Fr(I.)g(I.)g(Gol'dman,)e(Zh.)i(Eksp.)h(T)-8 b(eor.)33 b(Fiz.)e Fl(46)p Fr(,)h(1412)f(\(1964\))g([So)m(v.)i(Ph)m(ys.)h(JETP)f Fl(19)p Fr(,)139 3213 y(954)f(\(1964\)].)83 3388 y Fx(4)139 3424 y Fr(A.)54 b(I.)g(Nikisho)m(v)g(and)g(V.)g(I.)g(Ritus,)k(Zh.)c (Eksp.)h(T)-8 b(eor.)55 b(Fiz.)e Fl(46)p Fr(,)59 b(1768)53 b(\(1964\))139 3545 y([So)m(v.)33 b(Ph)m(ys.)i(JETP)e Fl(19)p Fr(,1191)f(\(1964\)].)83 3720 y Fx(5)139 3757 y Fr(N.)j(B.)g(Narozhn)m(y)-11 b(\025)-38 b(\020,)35 b(A.)h(I.)f(Nikisho)m(v,)g(and)g(V.)g(I.)g(Ritus,)g(Zh.)g(Eksp.)h(T)-8 b(eor.)36 b(Fiz.)e Fl(47)p Fr(,)139 3877 y(930)e(\(1964\))f([So)m(v.)j (Ph)m(ys.)g(JETP)g Fl(20)p Fr(,)e(622)g(\(1965\)].)83 4053 y Fx(6)139 4089 y Fr(J.)g(Bergou,)h(S.)g(V)-8 b(arr\023)-49 b(o,)32 b(J.)h(Ph)m(ys.)h(A)f Fl(13)p Fr(,)g(2823)e(\(1980\).)83 4264 y Fx(7)139 4300 y Fr(A.)57 b(I.)h(Nikisho)m(v)f(and)h(V.)g(I.)f (Ritus,)64 b(Zh.)57 b(Eksp.)i(T)-8 b(eor.)58 b(Fiz.)e Fl(46)p Fr(,)64 b(776)57 b(\(1964\))139 4421 y([So)m(v.)33 b(Ph)m(ys.)i(JETP)e Fl(19)p Fr(,)g(529)f(\(1964\)].)83 4596 y Fx(8)139 4633 y Fr(H.)g(R.)h(Reiss,)g(J.)f(Opt.)h(So)s(c.)g(Am.) f(B)g Fl(7)p Fr(,)h(574)f(\(1990\).)83 4808 y Fx(9)139 4844 y Fr(D.)g(P)-8 b(.)33 b(Cra)m(wford)g(and)g(H.)f(R.)h(Reiss,)f (Opt.)h(Express)i Fl(2)p Fr(,)e(289)e(\(1998\).)83 5020 y Fx(10)174 5056 y Fr(C.)48 b(Szymano)m(wski,)k(V.)c(V)m(\023)-46 b(eniard,)51 b(R.)d(T)-8 b(a)d(\177)-38 b(\020eb,)51 b(A.)d(Maquet,)k(and)c(C.)g(H.)g(Keitel,)139 5176 y(Ph)m(ys.)34 b(Rev.)f(A)g Fl(56)p Fr(,)g(3846)e(\(1997\).)83 5352 y Fx(11)174 5388 y Fr(C.)i(Szymano)m(wski)g(and)g(A.)g(Maquet,)h(Opt.)e (Express)j Fl(2)p Fr(,)e(262)f(\(1998\).)1653 5637 y(33)p eop %%Page: 34 34 34 33 bop 432 537 a Fx(12)523 573 y Fr(V.)31 b(P)-8 b(.)31 b(Y)-8 b(ak)m(o)m(vlev,)33 b(Zh.)d(Eksp.)j(T)-8 b(eor.)31 b(Fiz.)g Fl(49)p Fr(,)g(318)f(\(1965\))g([So)m(v.)i(Ph)m(ys.)h(JETP)f Fl(22)p Fr(,)487 693 y(223)g(\(1966\)].)432 860 y Fx(13)523 897 y Fr(C.)h(M)s(\177)-51 b(uller,)30 b(A.)j(B.)g(V)-8 b(oitkiv,)31 b(and)i(N.)g(Gr)s(\177)-51 b(un,)31 b(Ph)m(ys.)k(Rev.)e(A) g Fl(67)p Fr(,)f(063407)f(\(2003\).)432 1064 y Fx(14)523 1100 y Fr(C.)26 b(M)s(\177)-51 b(uller,)25 b(A.)h(B.)f(V)-8 b(oitkiv,)26 b(and)g(N.)g(Gr)s(\177)-51 b(un,)26 b(Ph)m(ys.)h(Rev.)f (Lett.)g Fl(91)p Fr(,)h(223601)e(\(2003\).)432 1267 y Fx(15)523 1303 y Fr(J.)g(D.)g(Bjork)m(en)i(and)e(S.)h(D.)f(Drell,)g Fp(R)-5 b(elativistic)28 b(Quantum)g(Me)-5 b(chanics)33 b Fr(\(MacGra)m(w-)487 1424 y(Hill,)d(New)k(Y)-8 b(ork,)33 b(1964\).)432 1591 y Fx(16)523 1627 y Fr(B.)f(Thaller,)g Fp(The)i(Dir)-5 b(ac)34 b(Equation)40 b Fr(\(Springer,)32 b(Berlin,)g(1992\).)432 1794 y Fx(17)523 1831 y Fr(R.)g(A.)h(Neville)e (and)i(F.)f(Rohrlic)m(h,)f(Ph)m(ys.)k(Rev.)e(D)f Fl(3)p Fr(,)h(1692)e(\(1971\).)432 1998 y Fx(18)523 2034 y Fr(J.)h(San)h (Rom\023)-49 b(an,)32 b(L.)g(Roso,)h(and)f(H.)h(R.)f(Reiss,)h(J.)g(Ph)m (ys.)h(B)f Fl(33)p Fr(,)g(1869)e(\(2000\).)432 2201 y Fx(19)523 2237 y Fr(J.)h(San)h(Rom\023)-49 b(an,)32 b(L.)g(Pla)5 b(ja,)32 b(and)h(L.)f(Roso,)h(Ph)m(ys.)h(Rev)f(A)g Fl(64)p Fr(,)g(063402)e(\(2001\).)432 2405 y Fx(20)523 2441 y Fr(J.)h(San)h(Rom\023)-49 b(an,)32 b(L.)g(Roso,)h(and)f(L.)h(Pla)5 b(ja,)32 b(J.)g(Ph)m(ys.)j(B)d Fl(36)p Fr(,)h(2253)f(\(2003\).)432 2608 y Fx(21)523 2644 y Fr(J.)g(San)h(Rom\023)-49 b(an,)32 b(L.)g(Roso,)h(and)f(L.)h(Pla)5 b(ja,)32 b(J.)g(Ph)m(ys.)j(B)d Fl(37)p Fr(,)h(435)f(\(2004\).)432 2812 y Fx(22)523 2848 y Fr(P)-8 b(.)33 b(Filip)s(o)m(wicz,)c(J.)k(Ph)m(ys.)i(A.)d Fl(18)p Fr(,)h(1675)f(\(1985\).)432 3015 y Fx(23)523 3051 y Fr(V.)g(B.)h(Berestetski)-11 b(\025)-38 b(\020,)33 b(E.)g(M.)g(Lifshitz,)e(and)i(L.)f(P)-8 b(.)33 b(Pitaevski)-11 b(\025)-38 b(\020,)32 b Fp(R)-5 b(elativistic)34 b(Quan-)487 3171 y(tum)i(The)-5 b(ory)41 b Fr(\(P)m(ergamon)32 b(Press,)i(Oxford,)f (1971\).)432 3339 y Fx(24)523 3375 y Fr(A.)f(Janner)i(and)e(T.)h (Janssen,)h(Ph)m(ysica)g Fl(60)p Fr(,)f(292)f(\(1972\).)432 3542 y Fx(25)523 3578 y Fr(M.)c(Reed)h(and)f(B.)g(Simon,)f Fp(Metho)-5 b(ds)30 b(of)h(mo)-5 b(dern)29 b(mathematic)-5 b(al)30 b(physics.)g(I.)g(F)-7 b(unc-)487 3699 y(tional)35 b(analysis)40 b Fr(\(Academic)32 b(Press,)i(New)g(Y)-8 b(ork-London,)32 b(1972\).)432 3866 y Fx(26)523 3902 y Fr(M.)48 b(Reed)h(and)g(B.)f(Simon,)j Fp(Metho)-5 b(ds)49 b(of)g(mo)-5 b(dern)48 b(mathematic)-5 b(al)49 b(physics.)f(III.)487 4022 y(Sc)-5 b(attering)35 b(the)-5 b(ory)41 b Fr(\(Academic)32 b(Press,)j(New)e(Y)-8 b(ork-London,)33 b(1979\).)432 4190 y Fx(27)523 4226 y Fr(M.)52 b(Reed)h(and)f(B.)g(Simon,)k Fp(Metho)-5 b(ds)52 b(of)h(mo)-5 b(dern)52 b(mathematic)-5 b(al)52 b(physics.)g(II.)487 4346 y(F)-7 b(ourier)59 b(analysis,)64 b(self-adjointness)i Fr(\(Academic)58 b(Press,)68 b(New)60 b(Y)-8 b(ork-London,)487 4467 y(1975\).)432 4634 y Fx(28)523 4670 y Fr(C.)31 b(Itzykson)h(and)e(J.-B.)h(Zub)s(er,)g Fp(Quantum)i(Field)f(The)-5 b(ory)38 b Fr(\(MacGra)m(w-Hill,)29 b(New)487 4790 y(Y)-8 b(ork,)33 b(1980\).)432 4958 y Fx(29)523 4994 y Fr(J.)47 b(W)-8 b(eidmann,)51 b Fp(Line)-5 b(ar)48 b(op)-5 b(er)g(ators)48 b(in)h(Hilb)-5 b(ert)49 b(sp)-5 b(ac)g(es)54 b Fr(\(Springer,)d(New)e(Y)-8 b(ork-)487 5114 y(Berlin,)32 b(1980\).)2002 5637 y(34)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0407291119751--